Firstly and almost tauologically, it REALLY depends; I don't feel that this information is enough for people to give accurate feedback.
What tier of grad schools are you aiming for? What are your GPA, research progress, and others like? Are you aiming for top schools only, or are you more open? What field are you interested in going into?
I am an international student, which means that I was practically never eligible for REU programs (most REU programs are funded by NSF and hence not open to international students). All my research experience was at my my undergrad institution. I did a total of 6 months of technician work in an experimental plasma physics lab and then a year of cosmological data analysis & analysis algorithm development work. I had no problem getting into some top grad schools. So, I would say that REU experiences, by themselves, do not mean much. Research at your undergrad school is completely fine, as far as you did meaningful research and your advisers are writing you strong letters. With that said, I did go to a R1 university and got letters from quite well-known professors, who all wrote me VERY strong letters.
So, no, REU themselves alone are not deal breakers/makers, but it does depend on what the other parts of your applications are saying.