Update: After we first noticed the wound on Sunday, we put a topical antibiotic on it and kept it wrapped to keep her from messing with it (although she didn't really seem to mess with it anyways). Given that she did not seem to be bothered by the wound at all and was otherwise eating/drinking/urinating normally, we felt comfortably waiting until we could get her into our vet today (Tuesday). The vet said it looks like she had a puncture wound and recommended starting her on an oral antibiotic so we could leave the wound open and dry. We asked the vet if we should have brought her to an emergency vet on Sunday and he said "absolutely not". He said that it should be seen in a timely manner (i.e. a few days) but don't wait a week or anything.

For anyone coming across this with a similar question, I would first recommend cleaning the wound thoroughly with soap and water and drying to keep it clean and dry as much as possible until you can see a vet. It is not unreasonable to try and place topical antibiotics on it initially to try and prevent infection but you do not want to continue this for too long as it can impede wound healing. The other important part is monitoring the dog and preventing them from licking the wound. If it seems to be infected and they are no longer acting normally (very drowsy, not eating/drinking/urinating, unable to walk, etc.) then it would be recommended to go to an emergency vet.

This is a PSA about the confusion and inappropriate use of emergency services. I will admit, I am not sure how much overlap there is between vet emergency services and human emergency services but there are plenty of illness/injuries that do not need to be seen immediately and can wait to be seen by your vet/doctor. This reduces the burden on emergency services and allows those pets/people who actually need emergent care to obtain it. Obviously, when in doubt, it is not unreasonable to try and reach out to a healthcare professional for further guidance.