Could I have a weak immune system? I just started working in day care and am constantly sick.

Common problem. You will probably get most of what the kids have that is easily passed through coughs, like URI's. If you are cautious & wash hands often you can reduce the germs passes by random surface contact. Once you have been in the business for a while your system will develop immune resistance & the illness frequency will drop.
Day care &sick. No you do not have a weak immune system.It is common as the kids get many different kind of viral and other infectons and the keep going around from kid to kid and you also get exposed to the same.You just have to take precautions with the kids by washing hands many times after coming in contact with the kids and not been to close to the babies who have cold etc.But sooner you will be immune to it.