This database from International IDEA makes it possible to find countries that offer people to vote in their home country elections even if they are abroad.
The Voting from Abroad Database is comprised of comparative data on external voting practices from around the world. Here you can further narrow your search and find detailed information on national electoral legislation from around the world that addresses external voting practices. The sources used to compile the data include the official web portal of governments, regional organizations working in the area of democracy and electoral processes, and research institutes specialized in the area of elections and politics. The database is continuously updated by researchers who perform periodic global studies. You can search for information alphabetically, by region or organization.