I am a foreigner working in the UK. I will need to renew my family's and my visas next year when our current visas expire. I am currently applying for jobs. I am thinking of asking my potential employers if they are willing to pay for my family's and my visa fees, something which I have never done, either when I first came to work here, or when I was working in another country before that. Is this something that companies would normally grant?
Yes, it is common, but by no means universal.
If they're the ones asking you to move overseas, visa support & costs are nearly always a part of the deal, and the family would typically be included in this case.
If you're applying for jobs locally, though, it's a bit of a question mark. The employer would certainly sponsor the visa, and would likely pay your fees (compared to your salary, the cost is likely small), but whether the family is included would be a matter of negotiation. And of course you'll be at a disadvantage compared another candidate who doesn't need a visa, not just because of the the cost, but because of the avoided hassle and uncertainty (what if the visa doesn't come through?).
Sponsorship for more expensive/optional endeavours like permanent residence or citizenship will also vary from company to company. When I applied for permanent residence in Australia, my employer at the time sponsored me and covered half the (sizable) bill, but with the stipulation that I'd have to repay it if I left within a year.
If I was your boss, and the company was small enough that I was able to make that decision and the company was profitable enough that the cost was insignificant, its a no brainer to pay for it.
Its much cheaper than paying the extra salary for you to pay it, as then there would be National Insurance, pension payments, income tax and so on all to pay.
Ask the question, make sure its not seen as a big issue, and if the answer is no, so be it, at least you asked.