If pipes are metal but not copper they must be galvanized iron pipes.
Check out this coupling at Home Depot.
Watts 3/8 in. x 1/4 in. Brass Female Compression x Compression Adapter-LF A157 at The Home Depot
If plastic tubing fits this adapter;
Remove existing stop valve and install galvanized tee. Install short nipple in each out port. Install a 1/4 turn FIP (female iron pipe) to 3/8" compression stop valve on each nipple. Reconnect faucet to one valve. Put adapter on the second valve. Connect plastic tubing to adapter. You will need a brass sleeve inside plastic tubing. You may have to buy a 1/4 coupling to get that sleeve.