Visitor Centre in Heathrow?

With hindsight: I would not have done it this way ☺
They sell Oyster Cards at the tube stations in Heathrow. You can buy them there prior to getting on a tube train. Buying ahead of time was totally superfluous.
And you say you are taking a "car service" (pre-booked taxi) from the airport. Why then do you need the Oyster Cards at Heathrow? You could equally buy them at the first tube station you come to when you first want to use the tube.

The other thing you need to know about Heathrow is it effectively 3 different airports: Terminals 1+2 (in the centre), T4 (south side) and T5 (west side), each with their own stations. It is many miles/Km between these.
Which terminal are you arriving in?

I found a webpage about Heathrow visitor centre: https://www.visitlondon.com/traveller-information/place/470024-heathrow-airport-visitor-centre

Located at Heathrow Airport, Heathrow Terminals 2 & 3 Underground station, Heathrow Airport Visitor Centre . . .

If you are landing at T4 or T5, you need to get a train to reach this visitor centre.

How much time should we budget to get to the Visitor Centre after claiming luggage, clearing cusoms, etc.? Assuming we will need to then backtrack from the underground station to arrivals to meet our car service.

It is not customs that takes time, it is immigration (passport control). Reckon 1 to 2 hours from landing to exiting through arrivals. Your "car service" will be monitoring your flight (presumably they asked for your flight number, so they will know which terminal to go to) and be waiting.
If you land a T2 or T3, then budget 30 minutes to go to the T2+3 tube station, do your business and get back.
If you land at T4 or T5, please rethink your plans.

We are arriving at Terminal 2. My twins are 10 years old and therefore eligible for free travel with a paying adult, but they are very tall for their age and are often mistaken for teenagers. Friends with a similar situation recommended ordering the Oyster 5-10 cards (which apparently must be applied for online and then picked up in person with proper id proving age) so that we don't get hassled about them riding for free since they do look much older (and we do not want to carry their passports with us everywhere). Of the visitor centres offered for pickup, Heathrow is the most convenient. The hired car is a benefit of our accommodation because we are arriving outside of their normal check-in time.

You are going to have to contact the car service to delay them compared with a normal pickup. This may cost you parking and other charges.

I doubt you would have had problems with your children travelling with you, although maybe they would like to go off on their own with Zip cards.

Jcm,
"My twins are 10 years old and therefore eligible for free travel with a paying adult, but they are very tall for their age and are often mistaken for teenagers." - It is not just a matter of being mistaken. The Tube has automatic ticket barriers to both get into and out of the stations. These barriers are opened with an Oyster Card. Without a card you can't get into or out of the system.

There is a list of Travel Centres where you can pick up the card here: https://tfl.gov.uk/travel-information/visiting-london/getting-around-london/visitor-centres
The last one on the list is:

Heathrow Heathrow Terminals 2 & 3 Underground station

At least you are arriving at T2.
I would advise your car company that you will be 30-45 minutes late. Perhaps ask for a phone number so you can phone them when you are ready to be picked up.

To be honest, I wouldn’t bother. You’re going to be tired after a transatlantic flight and a probably long wait going through immigration. Just take the car service from the arrival area after baggage reclaim.

Could they be challenged on age on the tube? Maybe, but I’ve never seen it in 6 trips to London. If it happens once, just get them regular Oyster cards and be done with it. The cost of transit is going to be a tiny fraction of a European vacation. And Heathrow is more of a city than an airport.