You can download MODIS and VIIRS hotspot data (as shown in Web Fire Mapper: https://firms.modaps.eosdis.nasa.gov/firemap/) from the FIRMS archive download tool:
https://firms.modaps.eosdis.nasa.gov/download/

Please note the MODIS archive is made up of standard quality data when available (approximately 2 months after overpass) and the near real-time data. The difference between the two is explained in the FAQs: https://earthdata.nasa.gov/firms-faq#ed-nrt-standard

It is recommended you review the FAQs before using the data and consult the relevant user guides for MODIS and VIIRS:
https://earthdata.nasa.gov/firms-faq#ed-user-guides

Please note: there is a limit on how much data you can download at once, so you may have to break up your request in to smaller requests - depending on the size of your area of interest and date range.

I hope this helps
Diane Davies
FIRMS User Support