Those who have dragons right now on the show:
Rhaenys (Meleys)
Daemon (Caraxes)
Rhaenyra (Syrax)
Jacaerys, or Jace (Vermax)
Lucerys, or Luke (Arrax)
Aegon II (Sunfyre)
Aemond (Vhagar)
Baela Velaryon (Moondancer)
Those who had dragons but who died or disappeared
Laenor Velaryon (Seasmoke; Laenor disappeared and did not seem to take his dragon with him, so we have no idea yet what happened to Seasmoke)
Viserys (Belarion, which died of old age years before he became king)
Laena Velaryon (Vhagar, now ridden by Aemond)
Helaena also had a dragon in the book at this point in her life named Dreamfyre, but we yet to hear of this dragon on the show.
At this point, given the current make-up of the Greens and the Blacks, here's the distribution of the dragons:
BELONGING TO THE GREENS (Alicent's and Aegon's side): Sunfyre, Vhagar, and likely Dreamfyre.
BELONGING TO THE BLACKS (Rhaenyra's side): Meleys, Caraxes, Syrax, Vermax, Arrax, Moondancer
Keep in mind, though, that Vhagar is by far the largest and more powerful dragon living.
Theoretically anyone with Targaryen blood (even if just their grandparent or great-grandparent was Targayen) should be able to bond with a dragon. At this point that means Rhaenyra's one remaining son with Ser Harwin strong, Joffrey Velaryon, should be able to bond with one, as should Rhaenyra's two children with Daemon: Aegon the younger and Viserys the younger. Rhaena Targaryen, Baela's twin, should also be able to claim a dragon. Helaena's three children also should theoretically be able to bond with dragons.