Internet over satellite is not really a novelty anymore and is offered by many service providers, many of them however only covering a certain area or region. Have you done any trivial and initial research to see if someone is offering what you need? Be aware though, that your budget may be stretched much more than you can imagine. The largest Globalstar plan with 75 MB (yes, that is MB) data is about US$200/month.

Are you planning to, typically, not reside in urbanised areas? If so, both electricity and internet might be a concern. Otherwise, it really doesn't need to be.

If you wait for a few years the answer is Starlink. Otherwise the answer as of 2020 is no - that's not possible.

Related: travel.stackexchange.com/questions/161384/…

You can get a BGAN terminal and an Inmarsat plan with 10/30GB of less-than-1mbps data a month for $2800/$4150 a month, or substantially cheaper service on a different plan just in Sub-Saharan Africa, so some things are possible, just not at all practical in terms of speed and cost. But if you can stick to areas with reasonable connectivity and explore remote regions on weekends and during vacation times, you have a lot more possibilities.

Or if you're staying in one place for a while, VSAT service could be an option, either your own or trying to make a deal with an internet cafe or other business that already has one.

Have you traveled much so far? It's pretty common to have decent internet even in guesthouses, and if you're staying a while you may be able to find a nice co-working environment with 24h access, and when you're away from that you can usually tether your phone if you don't have to deal with huge amounts of data.

The "developing world" is not somewhere completely devoid of technological access, India and most countries in East Asia and Southeast Asia (except DPRK, perhaps) have a functional cellular infrastructure, at least for population centres, sometimes even better than places like US, Canada or Australia. Unless of course, you are the adventurist type and plan to spend significant amount of time in the wild. But in that case, internet could be least of your concerns for your planning. Electricity and camping infrastructure would be a bigger problem.

Plan well about the countries you travel. Get a new SIM for data on landing which covers most of the country.

There are legal aspects to this question as well. India for certain has restrictions on satellite communications (iridium is not legal there). China bans most as well. Would need to check that your technology is allowed across your itenerary.

What do you mean by "India and parts of Asia"? India is already part of Asia. Beware that satellite phones are prohibited in India, and in some countries such as Turkmenistan there may be censorship that may complicate your work.

"I'm seeing two major obstacles here, electricity and internet." And of course a work permit, as well as figuring out how to pay taxes there, since you seem to want to work in a foreign country.

"I'm seeing two major obstacles here, electricity and internet" - I'd add safety. A first-world tourist with tons of expensive gear is a likely target for robbery. Especially in bigger cities.

"Sub-saharan Africa, India, and parts of Asia" will not all be on convenient timezones - In India, for example, you're looking at 10 hours difference from EST, and that's only going to get worse further East. Additionally, as @Szabolcs mentioned in their comment, you will need to be aware of the legality and tax implications of working in those countries.

You should clarify what you mean by "reasonable internet": are you wanting to check your emails and access a few websites, or log on to company VPNs, download large files, and stream video? As others point out, the former is easily possible via cellular networks, but the latter may be more of an issue.

@DaveMongoose India is worse than 10 hours, actually, because in the late fall and during the winter it's 10 hours and 30 minutes from EST, and during most of the year it's 9 hours and 30 minutes from EDT.

It would also be interesting if you clarified whether you intend to move to a place for a while before moving on to the next one, or if you are planning to move every single day or anything close to that.

Working remotely and going on holiday are not the same thing. I'd really reconsider your plans.