The expression is quite common in the US, but I think it's fairly recent - within my lifetime, anyway. (I've never used it.) It dates at least back to The Music Man- "Permit me, if you will, I'm Professor Harold Hill."
This use of will derives from the original meaning of Old English wyllan, which meant 'to want, to desire'. There are easily recognizable similar words that mean 'to want' in other Germanic languages: wollen (German), vilja (Swedish).
There are grammatical similarities as well. For example, in Swedish the plain infinitive (= infinitive without to) is us
Hi, CB, I agree that the sense of it is a polite invitation, similar to "if you please"; "if you wish"; "if you want to"; "if you would"; "if you don't mind." Cheers