I was wondering if the pudding mixes can be added to a scratch recipe to keep them moist. Like I have a vanilla scratch recipe that tends to be a little dry, can I add a box of pudding to the ingredients to help the with the dryness?

Yes, pudding mixes make the cakes moist plus they add a nice flavor to the cake. Yum!
A bit of sour cream can also moisten the cake. Your would need to play with its quantity to suit your "moisture preference" though.

Pudding and sour cream help, but make a test cake first. When I did this for the first time, my cake sank in the middle! Too much moisture I guess!

Great! Thanks so much guys