Not really. While you can, your landlord would win if your landlord acted promptly and the problem was abated. You are not entitled to an absolute non-interference with your right to posses the premises. While bed bugs would be quite annoying and I would not want to put up with that, your landlord has the right to attempt to remediate the problem.
If the problem did not go away, your real option would be to leave and your apartment ought to foot the moving expenses, of if the apartment has an alternative apartment, move to that one if it is not infested with insects.
Your question raises two different areas of law. Personal injury, as well as the Arizona Residential Landlord Tenant Act. There are very specific rights, remedies, and timelines laid out in the ARLTA. As a personal injury lawyer, I am not familiar with those deadlines but encourage you to look closely at them to see what rights you have.
As far as a claim for any personal injury, you would only have a claim if the bed bugs were caused by a negligent act on the part of the landlord. It does not sound like that is the case from the description you have provided. If the landlord knew of an infestation in the complex and did nothing about it in order to save money for extermination, there would be a different analysis. However, even if that could be proven, if you were lucky enough to get the problem treated and escape with only limited bites that quickly healed, even a claim under the law that could prevail would likely not be worthy of the effort and time involved due to the limited recovery.