Extract from the 1860 Census:
Name: Elijah Lambert
Age in 1860: 13
Estimated birth year: 1847
Birthplace: Vermont
Race: white
Sex: male
Relation to head-of-house: grandson
Living with: mother, and Francis and Rosele Bibo, ages 65 and 73, both b. Canada
Home in 1860: Winsted, McLeod, Minnesota
Post office: Rocky Run
Occupation: attending school
Siblings living at home: Almena (age 11) and Swedenborg (5)
Census place: Winsted, McLeod, Minnesota; Roll M653_571; Page: 361; Image: 366; Family History Library Film: 803571
Date: 8 Jun 1860

Extract from American Civil War Soldiers:
Name: Elijah Lambert
Enlistment date: 8 Feb 1865
Distinguished service: distinguished service
Side served: Union
State served: Minnesota
Unit numbers: 1147 1147
Service record: enlisted as a private on 8 February 1865
Enlisted in Company H, 1st Heavy Artillery Regiment Minnesota on 8 February 1865
Mustered out Company H, 1st Heavy Artillery Regiment Minnesota on 27 September 1865

Extract from the 1875 Minnesota State Census:
Name: Elijah Lambert
Age: 28
Estimated birth year: 1847
Birthplace: Vermont
Race: white
Gender: male
Residence county: McLeod
Residence state: Minnesota
Locality: Hutchinson
Father's birthplace: France [sic]
Mother's birthplace: France [sic]
Sibling living at home: Emanuel (age 20)
Census date: 1 May 1875
Line: 35
Roll: MNSC_11

Extract from the 1880 Census:
Name: Elijah Lambert
Age: 33
Estimated birth year: 1847
Birthplace: Vermont
Race: white
Sex: male
Relationship to head-of-household: head
Home in 1880: Township 125, Traverse, Minnesota
Marital status: single
Father's birthplace: Canada
Mother's birthplace: Canada
Occupation: farmer
Census place: Traverse, Minnesota; Roll: T9_635; Family History Film: 1254635; Page: 360.2000; Enumeration District: 2
Date: 7 Jun 1880

Extract from the 1895 Minnesota State Census:
Name: Elijah Lambert
Age: 48
Estimated birth year: 1847
Birthplace: Vermont
Gender: male
Race: white
County: Otter Tail
Locality: Fergus Falls Ward 4
State residency: 36 years
District residency: 8 years
Parents of foreign birth: yes
Occupation: farmer

Extract from the 1900 Census:
Name: Elijah Lambert
Age at last birthday: 53
Date of birth: May 1847 [sic]
Birthplace: Vermont
Race: white
Sex: male
Relationship to head-of-house: inmate
Home in 1900: Minneapolis Ward 12, Hennepin, Minnesota
Address: Minnesota Soldiers' Home
Marital status: single
Able to read, write and speak English: yes
Father's birthplace: Illinois
Mother's birthplace: Vermont
Occupation: none
Census place: Minneapolis Ward 12, Hennepin, Minnesota; Roll: T623 770; Page: 3B; Enumeration District: 151
Date: 5 Jun 1900

Extract from U.S. National Homes for Disabled Volunteer Soldiers, 1866-1938:
Name: Elijah Lambert
Estimated birth year: 1846
Birth state: Vermont
Admitted year: 1906
Age at admission: 60
State: South Dakota
County: Fall River
City: Hot Springs
Branch: Battle Mountain Sanitarium
Date of enlistment: 6 Feb 1865
Place of enlistment: Lebanon, Minnesota
Rank: private
Company: H
Regiment: 1st Minnesota Heavy Artillery
Date of discharge: 27 Sep 1865
Place of discharge: Nashville, Tennessee
Cause of discharge: or. War Department
Kind and degree of disability: lumbago; sciatica
When and where contracted: Arthur, Minnesota
Height: 6 feet 1 inch
Complexion: fair
Eyes: black
Hair: gray
Can read and write: yes
Religion: Protestant
Occupation: laborer
Residence subsequent to discharge: Fergus Falls, Minnesota
Married or single: single
Name and address of nearest relative: E S Lambert (brother), Fergus Falls, Minnesota
Rate of pension: $8.00

Extract from the 1910 Census:
Name: Elijah Lambert
Relationship to head of family: head
Age in 1910: 63
Estimated birth year: 1847
Birthplace: Vermont
Race: white
Sex: male
Home in 1910: Delaware Township, Leavenworth, Kansas
Address: National Home for Disabled Volunteer Soldiers
Marital status: single
Able to speak English: yes
Able to read and write: yes
Father's birthplace: Canada French
Mother's birthplace: Canada French
Occupation: farmer
Whether out of work on April 15, 1910: yes
Census place: Delaware Township, Leavenworth, Kansas; Series: T624; Roll: 444; Page: 62A; Enumeration District: 85; Part: 2; Line: 21
Date: 27 Apr 1910

Extract from U.S. National Homes for Disabled Volunteer Soldiers, 1866-1938:
Name: Elijah Lambert
Estimated birth year: 1846
Birth state: Vermont
Date of admission: 20 Nov 1906
Discharge date: 11 April 1907
Cause of discharge: transfer to Battle Mountain Sanitarium
Date of readmission: 17 Aug 1907
Discharge date: 5 Apr 1917
Cause of discharge: o. r.
Age at admission: 60
State: Kansas
County: Leavenworth
City: Leavenworth
Branch: Western Branch
Date of enlistment: 6 Feb 1865
Place of enlistment: Lebanon, Minnesota
Rank: private
Company: H
Regiment: 1st Minnesota Heavy Artillery
Date of discharge: 27 Sep 1865
Place of discharge: Nashville, Tennessee
Cause of discharge: or. War Department
Kind and degree of disability: lumbago; hemorrhoids
When and where contracted: Arthur, Minnesota
Height: 6 feet 1 inch
Complexion: fair
Eyes: black
Hair: gray
Can read and write: yes
Religion: Protestant
Occupation: laborer
Residence subsequent to discharge: Fergus Falls, Minnesota
Married or single: single
Name and address of nearest relative: E S Lambert (brother), Fergus Falls, Minnesota
Rate of pension: $8.00

Extract from the 1920 Census:
Name: Elijah Lambert
Age: 72
Estimated birth year: 1848
Birthplace: Vermont
Race: white
Sex: male
Home in 1920: Minneapolis Ward 12, Hennepin, Minnesota
Address: State Soldiers' Home
Marital status: single
Able to speak English: yes
Able to read and write: yes
Father's birthplace: Canada English
Mother's birthplace: Canada English
Occupation: farmer
Census place: Minneapolis Ward 12, Hennepin, Minnesota; Roll: T625_838; Page: 1A; Enumeration District: 230; Image: 648
Date: 2 Jan 1920

Extract from the 1930 Census:
Name: Elijah Lambert
Age: 83
Estimated birth year: 1847
Birthplace: Vermont
Race: white
Sex: male
Relation to head-of-house: inmate
Home in 1930: Minneapolis, Hennepin, Minnesota
Address: Minnesota State Soldiers Home
Marital status: single
Able to speak English: yes
Able to read and write: yes
Father's birthplace: Canada French
Mother's birthplace: Canada French
Military service: civil war
Owned a radio: no
Occupation: none
Census place: Minneapolis, Hennepin, Minnesota; Roll: 1099; Page: 3A; Enumeration District: 236; Image: 221.0
Date: 4 Apr 1930

He was in the Civil War, enlisting in the Union Army in 1865 when he was 18.
The Civil War muster roll from ancestry.com lists his middle initial as "H," his unit as "1st Minnesota Heavy Artillery," his rank at induction and discharge as "Private," the box number as 000546, the extraction number as 0005 and the record number as 0002562. The numbers refer to the microfilm copies in the National Archives. Union records were taken from microfilm roll RG94.
Ancestry.com lists his enlistment date as February 8, 1865, his unit as H Company of the 1st Minnesota Heavy Artillery Regiment, and his muster out date as September 27, 1865.
The 1st Minnesota Heavy Artillery Regiment was organized September 1, 1864, and mustered out September 27, 1865. It suffered no killed or wounded, although 87 enlisted died of accidents or disease.
Elijah Lambert died in the hospital of the Minnesota Soldiers Home, Minneapolis, Minnesota, age 89, and was buried 11 Jan 1937 in Fergus Falls, Minnesota.

Death notice from the "Star" of Minneapolis, Minnesota dated
Article 9 Jan 1937:
Lambert - Elijah, age 89. Services Monday, 10 a.m. at the Minn. Soldiers home. Arrangements by Joe Billman, 1400 Park av.