The company had already ceased accepting new AWS customers in Russia and Belarus.
Both Mariupol and Sumy are experiencing widespread blackouts following attacks by Russian forces.
The Mozilla Foundation is planning to give awards of up to $50,000 to each of the winning projects.
Thousands of tech workers left Belarus after the 2020 presidential elections and ensuing civil unrest means they are unlikely to return.
An internal memo does appear to offer some flexibility, however.
RT News and Sputnik News are no longer available for download from the App Store outside Russia.
Applications for the mentorship program open March 1.
Musk's pledge was decried by some as a publicity stunt, but Ukraine says the first SpaceX Starlink internet terminals have arrived.
The satellite communications giant says the outage is impacting internet services in Ukraine and other countries.
Microsoft also said it will not display any Russian state-sponsored RT and Sputnik content
The company has been "preparing for these circumstances" to "ensure no disruptions in the support and development" of its products.
Updated: The live tracking service is struggling under 10-20 times more traffic than normal.