Yes, the Election Administration and Voting Survey - conducted biennially since 2004 by the Election Assistance Commission - collects this data.
The 2016 report can be found here. In particular:
Approximately 80.1 percent of absentee ballots that were transmitted
to voters were returned and processed, with 1.4 percent of transmitted
ballots returned as undeliverable and 2 percent reported as spoiled
(e.g., the voter returned the ballot and asked for a replacement).
Ninety-nine percent of absentee ballots categorized as “returned and submitted for counting”
were ultimately counted in the 2016 election.

To put a raw figure on the number of ballots rejected, the report also states that "Nationally, by-mail voting constituted 23.7 percent of all votes cast in the 2016 election".
The FEC gives the total number of votes cast in 2016 as 136,669,276, so, according to the EAVS, there were approximately 32,390,000 by-mail ballots cast. As these, presumably, are the 99% which were counted, this means that approximately 327,000 by-mail ballots returned were rejected. As 19.9% of ballots sent to voters were not returned, we can calculate this figure as around 8,128,000, of which around 572,000 were returned as undeliverable, and 816,000 reported as spoiled.