I applied to Clinical Psychology programs this year, and I actually found that most schools I applied to "strongly preferred" that recommenders use their online form. So when you're filling out your app, you write in your recommenders' email addresses, and it sends them the form. The only thing is, this is very time-consuming for your recommenders, because each school has a different form, and sometimes there are technical difficulties. Different recommenders will feel differently about this, though; two of mine filled out all 14 online forms without complaint, but one of them said she didn't have time and opted to write a generic letter and upload it to interfolio.com. From there, I was able to send her recommendation to most of the schools on my list. However, about 3 of the 14 schools required the use of their online form, and would not accept recommendations sent in any other format.
Applying to 20 schools should be fine. You're expected to apply to 15 or so for Clinical Psych, and your professors will know that. Just supply them all your materials well in advance of the deadline (about 2 months is good).