Quick answer
Not for a relaxing first visit. Buses run all day between resorts like Caniço de Baixo and Funchal, Uber, Bolt and taxis are easy to find, and Funchal has plenty to walk to, plus the cable cars, the famous Monte toboggans and many tours round the island. If you want the freedom to explore trailheads, viewpoints and the north and west coasts on your own schedule, you definitely need a car.
Context
Funchal itself is walkable end-to-end, and SIGA runs frequent buses to Câmara de Lobos, Cabo Girão and Ribeira Brava. Bolt operates across greater Funchal at roughly half the price of a metered taxi.
Once you want to reach trailheads (Fanal, Caldeirão Verde, Pico do Areeiro), drive the ER101 around the north coast, or visit Porto Moniz, São Vicente and Calheta, a rental is by far the easiest option. Public buses serve these places but typically once or twice a day, which kills a hiking plan.
Roads are well-maintained but steep, narrow in old villages, and the VR1/VE1 expressways have lots of short tunnels. Diesel/petrol prices are EU-mainland normal.