FHA and Conventional loans do not require a home inspection. However, if the appraiser makes note of a deficiency, then the underwriter may require it to be fixed. This is at the discretion of the underwriter.
VA loans do require a "WDO" or pest inspection. Additionally, if there is a septic tank and/or well water, then inspections of those systems will be required also. Any deficiencies must be repaired before closing.
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Unless your buyer is getting an FHA or VA loan there is no requirement for the lender to inspect or request a copy of the inspection report. You're stuck on this transaction but I'd let the loan officer no this is the last client they'll ever get from you because their sticking their nose where it doesn't belong and that you and your clients have too many options to play games with them Lenders have been and continue to be somewhat out of control, it's why I never hitch my wagon or risk my livelihood to any one lender but routinely have 8-10 I can turn to and if one starts to act up, I fire them immediately and make sure that future buyers avoid them like the plague.
Let's remember that when the day is done we give them business and not the other way round, they can create as many rules and hurdles as they want, if I don't like them they that's the end of our business relationship.
I cannot imagine a Buyer not doing an inspection, and FHA and VA loans, both hinge on the condition of the property.
Even if it is not an FHA or VA lender, it does not surprise me that they would request an inspection; if only for the Health and Safety items.