Noun and verb
 

In Japanese we often hear the phrase お愿いします, here "愿い" is a noun here and then we add する after that to make it into a verb. Can anyone please tell me why we can't use the verb directly i.e. 愿う instead of 愿い so that we don't have to use する at all.

Do you mean "お願いします"?　I have never used the kanji "愿".

As you pointed out, "します"（"する"+”ます”) make verb from noun. But in this case, "願う" itself means "wish" not "ask a favor".

When you say "お金を貸してください。願う。" instead "お金を貸してください。お願いします。:Rent me some money. I beg your favor.", it does not work.

"お金を貸してください。私は願う。" will work as a foreigner's words though it is not natural.

"お願いします" is used to reinforce the words asking something. No Japanese says "お金を貸してくれることを願う".

No one knows the reason. It is just one of Japanese custom.