You could do Kona part of the time w/o a car but it would be limiting. The boats all leave from Honokohau harbor a few mins. drive north of town. Most shops have offices in town so you check in then meet at the boat.
One option would be to dive with Jack's Diving - they're on Alii Dr. on the south side of town. If you show up there, they will shuttle you to the boat. They also do guided shore dives in the afternoons.
There's several hotels within walking distance of Jack's - some would be too far to haul gear daily.
I think the Holiday Inn Expres is nearby and the King Kamehameha is a Courtyard by Marriott. It's actually behind the Kona/Kailua pier which is a shore dive - long walk from there to Jack's thouigh. There's also at least one operator with a smaller boat who picks up there - sorry don't know the name. There's also a
trolley that runs south along Alii from downtown to around Keahou - they'd probably frown on a lot of dive gear but I have seen snorkelers on it with mask/fins in a bag.
What you'd miss w/o a car is the Volcanoes NP, the waterfalls, the Waikoloa coast area, the observatories. The Big Island really is Big - I'd get a car. Pre-book - the walk-up Kona airport rates are a shock. I think we used Dollar last time.
Maui could work either way also but a car would likely be needed. You could stay in Lahaina, dive with one of 3 shops who keep boats in the harbor - walking distance from some of the smaller hotels - there and have access to all the food, etc. in the area. A lot of resorts are in Kaanapali and Kahana further north - you;'d need a ride for those - between properties is farther than I'd want to walk. The Sheraton at Black Rock is also a known shore dive - easy for beginners.
The Kihei boats all go to Molokini Crater or dive along the south Maui coast. Both suitable for newer divers. Probably stay off any trips that mention the back wall, it's often a live drop and about 400' to the bottom. The other beginner suitable dive would be the Cathedrals at Lanai - I did dive #12 there decades ago.
There's quite a few shuttles running around the island, tour buses to the various attractions etc. so if you planned it you could minimize the days you need a car. I was caught w/o the rental once so called a shuttle to pick me up at our condo in Wailea and went into Kihei for dinner - across the street was a beach and shroe dive. I think it was about $15 one way.
If you stay in south Maui, you'll need a car to get to the boats - they're at the Kihei Ramp a little north of town and there's not much nearby. Some of the resort/condo properties in South Maui are on named dive sites - there's several off Wailea Beach which is a decent dive. Polo Beach adjacent also but you'd need a car for the boats.
Being solo, there's options for escorted shore diving there but you often have to meet at the site.Shaka Divers, Maui Dreams Dive Co. are two in that area. In West Maui Extended Horizons does guided shore dives afternoons (they also own one of the boats) but they're not located near much except Mala Pier but not close enough to walk to it.They want you to check in at the shop then meet at the dive sites nearby.
Honestly it would just be a lot of work on either island without a car. Also the more affordable fares to Maui tend to go into the airport on the east side, all the diving is west. About a 45min. drive but there's hotel/airport shuttles.
Don't fly into Hilo instead of Kona - that's a 2 hr. drive across the island. There's no real diving near Hilo except at a couple shore sites. I don't believe there's even a dive boat on that side and only one shop that I know of - Nautilus.
my .02