Does VA or FHA have a requirement to provide notice of servicing transfers to the carrier/agent?

I am not sure I understand - what is a carrier/agent? You mean for insurance.

Yes, is there a requirement to notify the carrier/agent of the insurance company when a loan transfers for servicing purposes?

I believe it is customary and possibly a requirement of State law. I don't really have time to dig through all the servicing guides, but here is what RESPA has to say.

1024.33(d) Preemption of State laws. A lender who makes a mortgage loan or a servicer shall be considered to have complied with the provisions of any State law or regulation requiring notice to a borrower at the time of application for a loan or transfer of servicing of a loan if the lender or servicer complies with the requirements of this section. Any State law requiring notice to the borrower at the time of application or at the time of transfer of servicing of the loan is preempted, and there shall be no additional borrower disclosure requirements. Provisions of State law, such as those requiring additional notices to insurance companies or taxing authorities, are not preempted by section 6 of RESPA or this section, and this additional information may be added to a notice provided under this section, if permitted under State law.

In the FWIW column, the CFPB recently cited violations for force placing hazard insurance when coverage was already in place, and further for not correctly refunding charges for double coverage. My message is that it's in the servicers best interest to get it all settled the easy way.