So on Wednesday I decided enough was enough as far as waiting and contacted an office relatively close to me which I found in the trans-friendly listing on this site, Rockford Psychiatric Medical Services. I filled out the little "contact us" section with my email (I am pretty sure I didn't mistype my email) and asked some questions like what their experience was with trans clients and if their office is capable of referring for HRT, in addition to stating my desire to sort out personal problems. I contacted them on Wednesday and still have had no response. Am I being impatient and should just wait a little longer? Should I contact them again in case my email got lost in their inbox? Should I go searching somewhere else?

I would give them one more chance and contact them again. If they don't respond, at all, I've got a therapist that does online sessions and has written hundreds of letters recommending hormones, surgeries, gender marker changes, etc. Let me know how it turns out.

You should definitely find your own access to HRT.  Call around with different endocrinologists in your area and let them know you already are trans identified etc.  If you use your real name and pronouns since day one, they will almost definitely allow you prescriptions.  If you need therapy for a different reason, then find a therapist.  Don't expect any correlation between therapy/psychiatry and trans care.  They are two completely different things.  Best of luck.

I'll give the same advice I always tell my wife when it comes to leaving voice mails. "If it was important, they'll call back".

I would try again to contact them. Perhaps also via an end run in addition to their "Contact Us" fill in the blanks form. With a little digging there may be a direct email, or simple just call leaving a message that you'll like 5 minutes of their time to see if there may be a fit.

It's possible something happened to the message as well, I have confidence in a voicemail system's ability to deliver it, but not in the human listening to not accidentally hit the delete button when they put the phone to their face to listen :p

I would place one more call during business hours. But also shop around like they say, it can't hurt.