I just want to add to Buffy's answer, with which I agree.
Applying without recommendation letters, even if somehow possible is a signal of death :) Generally speaking, decent programs focus on research and want their students to do the same. Since a doctoral student is quite an investment to a department (time from professors, stipends, materials etc..), often they are fairly risk averse. This means that admission committees often (so I've heard, but it is a logically compelling argument) search for as many signals that a candidate can (a) pass the rigors of the coursework (e.g., through past GPA), (b) produce productive and methodologically rigorous research, and (c) not quit in the middle. The last three points can be conveyed through past research experience, a compelling statement, and letters of recommendation. These letters can help alleviate the aforementioned concerns by showing that a candidate has experience with conducting research, and that people (i.e., professors) that are proven as researchers believe that you can cut the mustard.
Long story short - get these letters. If you need, it is often not a year wasted to spend working for professors before applying. You can increase your chances at a good graduate program dramatically, and also get a better sense if this is indeed a path you with to partake in.