Maybe.

In Original letters, written during the reigns of Henry VI., Edward IV., and Richard III by Sir John Fenn and William Frere, there are multiple instances of people named Henry being referred to as "Harry". Most of these letters come from the late 1400s, within a few decades of the lives of the two kings, so we can say that using "Harry" as a familiar substitute for "Henry" would not have been uncommon at the time.

Some of the letters written by Henry IV and Henry V are collected in Original letters, illustrative of English history : ser.1-3 by Sir Henry Ellis. At no point in their letters do either king refer to themselves as "Harry". However, it should be kept in mind that these were official (and therefore formal) letters that would not have used nicknames or terms of endearment anyway.

In the plays, the name "Harry" is only used to refer to Henry IV in his own presence by his closest friends and worst enemies. Henry V, on the other hand, is referred to as "Harry" quite often to drive home Shakespeare's point that he is a "man of the people" who cares dearly about his countrymen regardless of social strata.

In the case of Henry V, Shakespeare uses an even more familiar nickname: "Hal". "Hal" is such an intimate name that it would have been the equivalent of calling Queen Victoria "Vicky". When Falstaff, Poins, Pistol, etc. refer to the young Prince Henry as "Hal", Shakespeare shows us his idealized monarch forming friendships with even the bottom rungs of English society.

Just because I can't find an example of Henry IV and V calling themselves "Harry" doesn't mean such documents don't exist. My own knowledge is more focused on Shakespeare's Histories than actual English history. If another of our users is aware of such an example, I would love to read it.

The Oxford English Dictionary lists it as a familiar version of Henry in use (at least) as far back as the late Fourteenth Century, which would land right in the middle of Henry IV's life (1367-1413). It is thus credible that the nickname could be used for Henry IV or V.

However, this does not itself constitute evidence that such a familiar name was used for either or (more to the point) that it was used in the way Shakespeare suggests. Holinshed contains no mention of "Harry" that I can find, nor does the Oxford DNB for Henry IV or V.

We do have evidence of Henry VIII referring to himself as "Harry," as another poster has pointed out. Given that Shakespeare grew up during the reign of Henry's daughter (and producing plays to please her), it's not beyond the realm of possibility that Shakespeare would conflate the two a tiny bit rhetorically.

Overall, I'd say it's plausible but unconfirmed.