We know for sure that Grey priest can't marry, but we do not know for sure for Blue priests.
That said, being able to marry and being able to have sexual relationships are different things.
We know for sure that sex is not forbidden in any way for priests, since Flower Offering is a thing.
.
My opinion is that Priests can't marry while they stay in the temple, for the simple reason that officially marrying would mean starting an household and recognizing the eventual child as heirs.
A priest, not being indipendent but being supported financially by the temple and its parent family donations, probably can't support a family. It doesn't make much sense for the temple to provide for his childs and wife, and even less the future generations after that.
It would also create the problem of how to classify his wife and childs. Would they be considered nobles or parts of the temple?
I think that in general Priests and Shrine maiden can't officially marry while in the temple, and every potential childs made would be sent to the orphanage as an orphan.
.
That said, I think that there is nothing stopping one priest to marry if he is able to return to the noble society. I doubt that there is any sort of oath of celibacy or anything similar.
As an example, if Ferdinand (who is able to return to the noble society) decides to marry, he would simply stop being a priest and start a family as a noble. Same would happen to all the ex-priests returned to noble society like Shikza or Christine.
Also, nothing stops priests from having love affairs. They just aren't able to marry and make things officially recognized.
That is also confirmed by we knowing that the former High Bishop had a mistress, and Delia wish to be able to become one since they get blue-priest like treatment (with their own attendants).