Hi @catcobb! I am an anthropology major, so we are in a similar boat as non-SLP applicants. I don't think it will hurt your chances at all, as long as you have a decent GPA and GRE scores and some experience. Not necessarily experience in CSD, but just extracurricular stuff in general. When writing my SOP, I made sure to relate the classes I have taken to SLP and how they would make me a better clinician in the future.
I didn't have professors that I knew really well, but found a professor I wanted to ask. I didn't just want to stop in and put her on the spot or catch her off guard so I sent an email too. I basically said that I wanted to schedule a meeting to discuss if she would be willing to write me a letter of recommendation. I said I was going to stop in, but I knew how busy the start of a semester can be and I didn't want to catch her at a bad time or put her on the spot. I also said that I had a packet prepared that included the schools I am applying to and why, a draft of my personal statement, my resume, and academic transcripts/information and if she was willing to schedule a meeting what times/days worked best for her. She emailed me back the next day saying she was happy to and that I could just send in the information electronically and didn't even need to meet with her. I thought it would have been weird to send an email saying that I wanted to meet with her, but not say what it was about. A lot of professors know that this is the time when students are asking for LOR, so they are expecting it anyway.
Hopefully this helps!