Richard,

Simply mention it to the movers that you contact or that contact you. Make sure you explain in detail how the move will be paid for, since most movers will accept public assistance checks, but its important to make sure they understand the procedure.

You can start here: get free moving quotes, and in comments specify that the move will be paid for with public assistance funds.

Just to note: many movers actually like working with clients whos move will be paid for by a third party (e.g. goverment, employer)