we do.
but it's not a 'free internet' as we use today.
while it's 'dos-based' internet as an early one. (My daddy had an experience using ones). it is correctly 'intranet'. a WAN owned, administrated, and moderated by an organization that owns the network.
right now i think there's THREE organization in the post-war FO settings that have ones.
1. Enclave; being successor to the US Government, and given that Vault Tec is not a pure private organization but it's actually 'State Enterprise' (just like AMTRAK). by the FO2. it is generally accepted that Vault Tec is now integrated into the Enclave.
before ZAX stationed in the Raven's rock created John Henry Eden' personality. ZAX was tightly linked to the Poseidon Oil Rig (which housed the Enclave until its destruction at the end of FO2) and somehow, unflinchingly served its creator.
2. Brotherhood of Steel; another successor of the US Army. and it is said to inherit hi-tech stuffs (and attempting to develop new ones or improve the existing tech they have). During Owen Lyon's mission in the east coast, to be specific, looking for the Pentagon. Lyon's warband maintained communication channel with the Lost Hills throughout the operations. and once Lyon's warband secured the Pentagon, Lyon reported the warband's findings to the Lost Hills. finally the Lost Hills granted Lyons the rights to found an east coast chapter. this tells us that the Brotherhood has some form of internet.
3. Mr. House. the founder of RobCo, and the keeper of the advanced computer technology of the FO settings. if he doesn't have an Internet, how will the two factions above have access to it?