Has the Republican party ever asked for a recount on presidential votes? If so, do you have any examples?
The answer appears to be yes. Contemporary newspaper accounts suggest that state Republicans requested recounts on the vote in New Hampshire in 1916, which Woodrow Wilson apparently won by 56 votes, one of the smaller absolute margins in a presidential election.
It isn't official yet, but it seems to happen in 2020 at least in Wisconsin:
The Trump campaign says it will “immediately” request a recount in the battleground of Wisconsin. But under Wisconsin law, a campaign can’t petition for said recount until the Wisconsin Election Commission completes the canvass from county election boards.
“Despite ridiculous public polling used as a voter suppression tactic, Wisconsin has been a razor thin race as we always knew that it would be. There have been reports of irregularities in several Wisconsin counties which raise serious doubts about the validity of the results. The President is well within the threshold to request a recount and we will immediately do so,” Bill Stepien, Trump campaign manager, said in a statement.