Taxi for 5 people from Heathrow Airport to London Central

Probably but you might be better off booking a car service. They meet you as you exit customs and drop you off in front of your hotel. Unlike a black cab, you know the price.

In a black cab, three people sit on the bench facing forward and two sit on the jump seats facing rearward.

There is of course the option of the tube as well. Piccadilly line from LHR to Victoria with a cross platform change at Hammersmith. The tube starts at LHR so no issue with seats. £5.60pp and almost certainly quicker than any road transport.

5 adults? I don’t think black cabs have much of a trunk. So the bulk of the luggage would have to be in the passenger compartment with you. Probably too snug for the ride into London. There’s a bus from Heathrow to Victoria coach station which is adjacent to the rail station. Walk or taxi to your lodging. Or as mentioned above take a minicab. I have used blackberry cars to get to Heathrow. Ride was fine.

https://www.blackberrycars.com/

I recently did a trip with five adults plus luggage in a black cab, and it was fine. The driver can take a bag in the front compartment, and you can fit bags under the fold down seats as well. Tight but perfectly do-able.

Black cabs can take more luggage than you think.. I have before now taken a Black Cab from Waterloo (when Eurostar used Waterloo) to Euston. Two of us, three bags, 2 fully assembled bicycles in the passenger compartment. No Probs.

Simple answer to the OP’s question is yes.

Can I just go to the taxi stand and expect them to take 5 people with carry-on luggage?

Yes. Five people with carry-on luggage will fit easily into a black cab and you can walk to the taxi rank and take the first available cab. However, it will be expensive so if costs are a concern then a pre-booked car service will be a better option.

Yes, 5 people with carry on luggage can fit into a black cab. They will be lined up waiting for you at the taxi rank.

Cheaper and far more comfortable would be the coach (bus) from Heathrow to Victoria Coach Station.
We arrived at Terminal 5 and followed the signs to the bus/coach station. The buses/coaches pull into the stalls right outside the sliding glass door at T5 coach station Heathrow.
The coach you want will say "London Victoria" on the header over the front window.

Buy your ticket after getting on the coach from the driver, or from the ticket machines inside the coach terminal. (a small room with coffee stand, restrooms, and benches).
Comfy, clean seats on the coach (more comfy than seats on most planes), WiFi on board. You ride up high so you can see the roadway and neighborhoods as you travel into London.
This coach is NOT in any way like the run-down buses in the USA. In England, they are brand new and comfy.

Price for the journey....about 6 pounds each person. Could have gone up a little since I traveled last.
Take carry on bags onboard with you.
Any large roller bags the driver will put them in the luggage hold under the bus/coach.

When you arrive in London, my guess is you'll be walking distance to your hotel.
The Victoria Coach Station is right across the street and down a bit from the Victoria Train Station.

This coach is used often by some members of this forum, especially those who like to stay at favorite hotels in the Victoria area.

Thank you all. Sounds like a taxi or the bus will work perfectly!