Individual Census Records from 1790 to 1950 are maintained by the National Archives and Records Administration, not the U.S. Census Bureau.

Indiana did not become a state until April 29, 1816, so the earliest federal census will be the 1807 Territorial Census currently found on Ancestry.com by subscription. The 1820 census started August 7 the first Monday in August 1820 from What day was the census taken each decade? at the United States Census Bureau.

August 29, 2017 post by Indiana Historical Bureau on Facebook:

On August 29, 1814, the Indiana Territorial Legislature adopted a resolution requiring a census of the "free inhabitants" of the territory, which excluded Native Americans and enslaved African Americans. In 1812, the U.S. Congress recommended that the territory's population be at least 35,000 as a qualification for statehood. The census, completed in 1815, enumerated the population of Indiana at 63,897. Learn more about Indiana's road to Indiana Statehood.

Allen County, Indiana was organized December 17, 1823 from Delaware and Randolph counties; and officially formed as a county April 1, 1824, so there are no Federal Census records for Allen County, Indiana in 1820. Allen County residents might be found in either Randolph County NARA Reel 13 or Delaware NARA Reel 14 in the 1820 Federal Census. Contact Allen INGenWeb if you find any future Allen County, Indiana residents on either reel.

1820 Overview at United States Census Bureau.

Census Instructions

The decennial census has always required a large workforce to visit and collect data from households. Between 1790 to 1870, the duty of collecting census data fell upon the U.S. Marshals. A March 3, 1879 act replaced the U.S. Marshals with specially hired and trained census-takers to conduct the 1880 and subsequent censuses.

During the early censuses, U.S. Marshalls received little training or instruction on how to collect census data. In fact, it was not until 1830 that marshals even received printed shedules on which to record households' responses. The marshals often received limited instruction from the census acts passed prior to each census. 

Beginning with the 1880 census, specially hired and trained census-takers replaced the U.S. marshals. Door-to-door census by temporary census-takers was the primary method of conducting the census until the U.S. Census Bureau began mailing questionnaires to households in 1960.

As more and more households received and returned their questionnaires by mail, the role of census-taker changed. Today, the majority of households are counted by mailed questionnaires. Census-takers visit places frequented by transient households (shelters and soup kitchens, campsites, etc.) and households that do not return their mailed questionnaires (during the "Nonresponse Follow-Up" phase of the census). As a result, the "Instructions to Enumerators" provided here include the congressional acts U.S. marshalls reviewed during the early census, specially-published instructions for door-to-door census, and lastly, guides used for the limited number of personal interviews conducted during nonresponse follow-up operations.

Copied from the Census Intructions at the United States Census Bureauthat show 1790 Instructions, but not again until 1850 Instructions through 2010.

Page 322 starts Randolph County on Reel 13 - Population schedules of the fourth census of the United States, 1820, Indiana [microform] (1959) at Internet Archive

Page 16A starts Delaware County on Reel 14 - Population schedules of the fourth census of the United States, 1820, Indiana [microform] (1959) at Internet Archive

Links to Online Census Records

1. Ancestry.comblank 1820 census form
2. FamilySearch.orgWiki 1820 Census Information
3. FamilySearch.orgbrowse 1820 Indiana Census by County
4. FamilySearch.orgSearch 1820 U.S. Census Index
5. FamilySearch.orgWiki for Indiana Census 1807 - 1940 links organized by year to various online census records.
6. Internet Archive1820 census on microfilm 3 rolls for Indiana, no index, digitized from NARA microfilm.
7. Rootsweb.com 1820 Indiana Federal Census Index for the state
8. Use Steve Morse Census Search Tool for any census