It is a mix.
Once FEMA gets involved and people are being brought in from outside it is mostly paid work. Volunteers are typically local, and FEMA has local hire positions which may convert some (many?) of those volunteer workers to temporary paid positions. FEMA also has experienced part time workers that get brought on to work disasters. FEMA doesn't leave as soon as the news moves on, there will be a FEMA presence sometimes for years afterwards to help with the rebuilding, helping with insurance claims, helping the local agencies contract long term repairs etc. I worked with some FEMA workers in 2021 during Covid, and some of those "part time workers" can spend 3-6 months at a large incident like a hurricane or earthquake. When they time out a new batch comes in. FEMA has a tiny full time work force, but a huge part time work force.
FEMA is mostly administrative, they organize the use of other responders, and subject matter experts. When the news reports FEMA is responding that is a huge simplification. FEMA's #1 job is organizing and writing checks, many other agencies are involved in the on the ground work.
There are 15 Emergency Support Functions under a FEMA response each of these falling under another Federal agency with the appropriate experience. Some examples ESF 1 is transportation, with a focus on getting local transportation up and running again. This is managed by the US Department of Transportation, ESF 3 is Public Works, which is managed by the US Army Corp of Engineers, ESF 4 Firefighting, is managed by the US Forest Service since they have experience managing huge wildfires, ESF 8 Public health is managed by the Department of Health and Human Services, ESF 10 is oil and hazardous materials, which gets managed by the EPA and US Coast Guard, ESF 13 Public safety / security is managed by the Department of Justice.
Under this you will have state and local agency personnel, utility companies, contractors etc which come from all over and will work under the different ESFs, managed by the agency with authority for that function.
These are very general areas, when I have responded to FEMA incidents as a US Forest Service Firefighter I've never had anything to do with firefighting, it has mostly been in a logistical function.
It is simply not feasible for most people to volunteer extended periods of time. If local they likely have their own issues to deal with, and if from outside the impacted area how many people can give up weeks and months of their time, time that they probably need to be employed.
Most states and counties / parishes / townships etc have similar agencies to FEMA, often some variation on Office of Emergency Services or Civil Defense way too many for anyone to track beyond their local area, but most work somewhat like FEMA just on a smaller scale.
Responding to a major disaster is very different from volunteer firefighters and emergency medical workers (ambulance) where the time spent volunteering is relatively small blocks of time, a few hours here, a few hours there, maybe an occasional all nighter or even a few dayers when there is a major incident, but those are fairly rare occasions.
Once a disaster goes beyond the local jurisdiction, there are often means to provide compensation to volunteers. When a fire burns onto National Forest, the USFS will pay the volunteer firefighters an agreed on hourly rate for the time spent on the incident from the point that the fire crossed onto the FRA (Federal Response Area). CALFIRE has a similar policy in California for fires that burn onto SRA (State Responsibility Area).
