You are correct, ideally you want two professors you conducted research well to write you a rec letter, after all, you are applying to a RESEARCH internship. These programs typically have very low acceptance rates and competition is intense. It is uncommon to see people without research experience to land a spot, unless these programs are meant to give people from schools without a lot of resources the opportunity to conduct research. Also, the thing about having no experience is that your personal statement, which is supposed to be an essay where you explain your background and preparation to be successful in the program, will be all over the place. Another thing to note is that a lot of these programs are NOT meant for people that want to do a medical degree alone, so make sure you check on the program requirements--they are mainly targeted for students wanting a PhD or combined degrees. Also, make sure you're not only applying because it'll make a difference in your med school apps, the adcom can usually see this very easily.
PS: Non-research related ECs won't really help you land a research fellowship.