You should be alright. I believe all counties except for maybe Pope have tags left. Unfortunately the IDNR does not have a very good inquiry system and won't have your information in there for a couple weeks. If you applied for the tag you should be set. The best bet is to see how many tags remain for the county you are applying for now and next time they update the amount of tags as long as its not zero you should be ok. The inquiry system though is your best bet, I'd check it every couple of days.
Personally, I wouldn't worry. As long as you make it in before the tags go over the counter you should be set. What county were you applying for?