They do get their own equipment. If you look, you can see them unplugging keyboards, mice and headsets after matches. Every player gets to pick whatever gear they want. Riot doesn't buy their standard gear, but does carry reserve gear in the event that a player's gear breaks on stage. Each org is responsible for their own players' gear. The rules each org has vary from team to team. A team sponsored by a manufacturer of gear may have a contract with the manufacturer involving what gear players use on stage. A team sponsored by Logitech might force players who prefer a Razer mouse to cover the Razer logo on it, for example.
Depending on what gear a player wants and what team a player is on, a player might be forced to buy their gear out of their own pocket, or it may be paid for by the team, or the org might contribute part and the player has to pay the balance.
The only rules Riot has on gear are: 1) No gear may interfere with another player's computer or gear; 2) No gear that requires a program to be installed on Riot PCs may be used; 3) No gear may provide an unfair advantage to its user or team (exceptions may be made for gear that compensates for a disability)