Most agencies operate off of annual contracts put in place well before a storm hits. This is handled with the individual agency (city, county, etc.). The contractor getting hired and/or paid is handled by that agency, not FEMA. FEMA does not hire or directly pay contractors.
Like others said, it takes a BIG outfit to be a prime for a jurisdiction of any size. The contractor has to be able to cash flow millions between payments. Payments would most likely be monthly at best.