After the Constitutional Convention had been going on for about a month with slow progress, Benjamin Franklin famously suggested that "prayers imploring the assistance of Heaven, and its blessings on our deliberations, be held in this Assembly every morning before we proceed to business". Several objections were raised, and it seems that the one that carried the day was that the convention had no funds to hire a chaplain.
Why was it taken for granted that the convention would have to hire a chaplain in order to have prayers, as opposed to e.g. calling for volunteers from among the delegates, having Washington (the president of the convention) call on a delegate daily, or drawing up a rotation? Did the predominant religion(s) at that time and place not allow non-professionals to pray in public?
The delegates to the Constitutional Convention were very practically- and politically-minded. Although some of them were religious in their daily lives, the debates at the Constitutional Convention were mostly hard-headed discussions of power, representation, and money. Basically, they didn't bring in a chaplain because they didn't want one.
The historian Richard Beeman, whose book Plain, Honest Men: The Making of the American Constitution (2009) gives a chronological, blow-by-blow account of many of the major debates, devotes several pages to this episode. He suggests that the other delegates didn't take Franklin's suggestion about prayer very seriously, but nobody wanted to hurt his feelings by saying so.
Franklin's motion calling for prayers was seconded by Roger Sherman of Connecticut, and then, in Beeman's words, "[a]n embarrassed silence followed." Alexander Hamilton came up with the excuse that, if word got out that they were asking for prayers, the public might believe that they were desperate.
Hugh Williamson of North Carolina came up with the "gosh, we just have no funds to hire a clergyman" answer. Beeman notes that this idea is implausible on the face of it. After all, there were no doubt plenty of clergymen in Philadelphia who would have been willing to show up and lead as many prayers as they wanted for free.
Edmund Randolph of Virginia, trying to help Franklin out by coming up with a compromise, then suggested that the Convention ask "that a sermon be preached at the request of the Convention on the 4th of July . . . and thenceforward prayers be used in the Convention every morning." Franklin seconded this motion, but it also failed.
Afterward, Franklin himself explained his understanding of what had happened. He wrote at the bottom of his notes, "The convention, except three or four persons, thought prayers unnecessary!"
My own take on religion in early America (this part is not from Beeman) is that there was nothing in the dominant forms of Christianity that would have forbidden the delegates from leading prayer out loud. However, they would have found it odd to do so, and I can hardly imagine them doing it. You might think of it in the same sense as, "If one of us broke his arm, we wouldn't try to splint it ourselves, even if some of us had first aid training. We are in the middle of Philadelphia -- of course we would call for a doctor!" If they had wanted to have prayers, they would have regarded calling in a chaplain as the logical thing to do.
Source: Richard Beeman, Plain Honest Men: The Making of the American Constitution, pp. 177-181.