Proceed, but make sure you have an inspection contingency.

Do I ask for an inspection contingency in the purchase agreement? 

@Duane Johnson Yes, that's exactly what happened they offered more than asking to cover their closing costs and waived inspection. 

Does a 5-day inspection period seem reasonable for a 2 to 3-unit property in Chicago?

The property was in bad shape and we had to get an extra inspection to get the deal done but it's possible.

My agent can find the deal but I still do my due dilligence, analyze numbers and get an independent inspections.   

The home has to be in move-in ready condition to pass an FHA Appraiser's inspection.

My question is do I need a permit and inspection to begin building? 

You can USUALLY include an appraisal or inspection contingency.

I've tried various things: shorter inspection time period (typically it's 10 days here, and if you do a 3 or 5 day inspection time you can be attractive b/c the seller can get it back on market quicker if you find something you don't like during inspection period); non-refundable earnest money (maybe $1000 of money that you will not get back for any reason on your side - not if the seller breaches-- should the offer not go to completion - this shows you are an extremely interested buyer, but you could lose the money, so use cautiously and with the advice of your agent who knows how to correctly write that into the contract); put a limit on the repair request no required by lender - cap it at $0/as-is, or $500; a higher earnest money - like $5000, shows you are serious.