From my previous experiences, it depends on which internships you are talking about. People get in with or without previous research experiences, and there are other factors too, such as where you went to undergrad, your essays in the application, minority backgrounds, etc.
The only thing I could tell you confidently is that getting into these programs is competitive, so I would apply as many internships as you can if you genuinely find any interest in their programs and/or see yourself doing research through them. I saw someone who compiled the data and made a list of internship programs with the percentage of acceptance, and if I remember correctly, these percentages were very low.
(Here are my anecdotal examples.
1) I have a few friends of mine from minority backgrounds who did the first summer research at a local undergrad, continued their research during academic year(s), and got into HHMI Exrop in the following summer(s).
2) One of my family members had zero research experience and mediocre GPAs, but applied to 20+ internships and got into 2 internships for this summer, one at Mayo and the other through NSF REU.)