No. Two extra letters per household (one ballot, then one returned ballot), largely spread out over days or weeks, is just not that much more volume for the USPS. That's of course the maximum; in practice the average per household will be much lower than that. Further, as stated elsewhere, non-package mail volume has collapsed because of COVID19.
What does matter is the backend, the infrastructure for counties and states that previously did not use mail ballots, or only used them for absentee voting, to a) properly send them out to the right recipients, then b) accept and process them. All this a) of course has to be done by early October to give voters a month before election day, and b) has nothing to do with the USPS.