Minicab to train station from Heathrow?

To get in and out of tube stations in London expect a lot of walking...since one of your sisters is using a cane I would avoid the tube as there are lots of stairs and long, crowded tunnels that don't go hand-in-hand with someone who may be on the slower side of things. If you do decide to take the tube, here is a list of tube stations that have elevatorsAvoiding the Stairs Tube Guide.
A taxi or minicab will cost you. It will also take quite a while to get into central London depending, of course, on which train station you are trying to get to. Plan on an hour at the very least, 2 or 3 if traffic is bad. Last summer it took almost 3 hours by car to get from Heathrow to east London...never again!
Have you considered taking Heathrow Express to Paddington Station? It's easy and quick but a little expensive - might be a good option though if you just want to get into town fast. It's a 15 - 20 minute ride. From Paddington it's only about a 15 minute tube ride to St. Pancras. From St Pancras it's about 60 mins to Canterbury and from Charing Cross, Victoria or Waterloo East it's about an hour and a half...they all will get you to Canterbury but some will take you to Canterbury East and others will take you to Canterbury West - not sure which one you need.
As for customs, it can take as little as 30 minutes if it is uncrowded or longer if it's busy. I would leave at least a two hour buffer between your scheduled landing time and the train you want to catch...a little longer maybe just to factor in a late flight or crowds.
Have a great trip!

I'd recommend a cab or a car service hire, which means you book a car that takes you into the city. (Check elsewhere around here for details and recommendations. Basically, you tell the service how many people, how much luggage, when to meet you, and where you need to go. As you leave passport control and customs, your driver will be one of a crowd of people and will be holding up a sign with your name on it.)

Cabs and car service cost roughly the same, but split 6 ways shouldn't be terrible. The drive in takes about an hour. A car service would send a van for 6 people plus luggage.

Don't mess with the Tube after a long flight, and looking at a two-hour trip to Canterbury. You'll be tired, and it's a pain hauling luggage around the Tube.

It's Passport Control where you will queue up and wait your turn. Allow an hour so so. Might very easily take less time, but could also take more if you arrive within minutes of a batch of other flights. After clearing passport control, most folks walk through the "nothing to declare" door at customs, under the watchful eye of customs officers.

I recall a long walk from Canterbury station into the town center, so a cab may be in order there. (Canterbury has two stations that are pretty close to each other.)

A car service will be much cheaper than a taxi. Many people on this board recommend Just Airports. You'll need at least an MPV (minivan) - or maybe an MPV plus a Saloon (sedan). There is a discount for paying cash. The URL is justairports.com.

I see that you have a rail pass. However, you may still want to consider a taxi direct to Canterbury as a start to your trip. At around £100 divided by 6 people it is not that extravagant, and has a lot of convenience, not having to go into London and taking you right were you are staying. If you were saving 6 train fares it might have been more advantageous.

In my previous post I said that a car service would be cheaper than an taxi. It also will be cheaper than a minicab.

St. Pancras is in postal code N1. The cash fare to anywhere in that postal code is £49 for an MPV and £37 for a Saloon.