Since the question is about a purchase from a private individual the answer is no.
If you were in MS on PCS or TCS orders your state of residence for purspose of the GCA would likely be MS. However since you are on TDY orders your state of residence is IL (your permanent duty station) not MS and therefore you may not purchase from a private individual.
I may be off but the line between PCS/TCS and TDY is 179 days. I don't have time to research the issue right now.
You might be able to purchase a long gun from an FFL depending on MS and IL law.
Originally posted by Don P
I may be wrong here. Its my understanding that anyone that can legally own a long gun can purchase a long gun anywhere he or she may be no matter the state they are in.
From an FFL but not from a private individual. Any purchase of a long gun between individuals who are residents of two separate states must go through an FFL in either the seller's or purchaser's state depending on state law. For the purchase of a handgun between such individuals would have to go through an FFL in purchaser's state.