This is technically possible using the variables MIGRATE1, STATEFIP, COUNTYFIP, MIGPLAC1, MIGCOUNTY1, and RENT. There are a number of limitations you should be aware of, though:
About 40% of respondents are in geographic areas that cannot be identified to the county level, because the county has too small of population. These are identified by COUNTYFIP==0 or MIGCOUNTY1==0. For the current residence, the lowest geography is PUMA, which often comprise multiple counties. For last year, the lowest geography is MIGPUMA1, which is a coarser geography.
The sample sizes per county (for identifiable counties) range from 769 to 101,631 in the 2017 ACS. You could combine multiple years to increase the sample, or use a 5-year ACS sample.
If you look only at individuals in identifiable counties who moved from other identifiable counties, the sample sizes range from 33 up to 7360.
About 25% of respondents in the sample paid any rent.
Another thing to consider is to use just the MIGRATE1 variable. This indicates whether a respondent resides in the same house as one year previously, or not. This would eliminate the need to consider last year’s county.