Head coach as chair?

Probably, I guess it depends on the rules of a particular club and the job requirements.

Is it important?

N&G is correct, that officially you would have to check the rules.
However, I think it poor practice, and would check quietly with the sport's governing body, although the exact relationship between the governing body and local clubs can be very variable.
I think it poor practice because there could be both conflicts of interest and an absence of the usual checks and balances.
However, it can be so difficult to get people to do these jobs now, that it may be that a coach with an interest in keeping the club going is the only person who wants to stand!
If that's the case I would be influenced by how 'important' this club is - a few adults who enjoy getting together is a different kettle of fish from one that trains youngsters and enters them into representative competitions.

I would try to find someone else to stand, and if you can't, suggest that the coach becomes a 'facilitator' rather than a formal chair.
Sometimes, sadly, clubs do fold.

It seems a conflict of interests to me. Better perhaps to have some diversity of roles within the club.

Do not feel there is anything implicitly wrons with this. Is he paid as Head Coach, or is it a totally voluntary role. If he is employed by the Cttee, then probably he could not Chair the cttee that employs him. but if he/she works on a totally voluntary basis, not so much of a problem. Personally, the to roles which I feel must be kept separate is that of Welfare officer and Head Coach.
Is this sports club mainly for children or for adults?

The sports club is a ski club for those with special needs, children and adults. All coaches are volunteers. You are right. It is so hard to get people to sit on the committee and actively take part. The head coach and I are practically running it. I will be made secretary at the next AGM. The chair is stepping down and the person who wants to be chair is a bit hot headed. So we’re not sure what to do.

It depends what you fear they will do.

Put some sort of guideline in place that the hot headed joker can make no decisions at all without consulting the committee for approval.

If you feel this person may lose their temper with clients/children then your rules should reflect the will of the committee. eg. ONE warning only and then out.

It doesn't sound as if there are any alternatives to electing this booby, so this is the only thing you can do. And put it in writing.

Shouldn't the committee and the chair be elected by the members of the club? Do you have a constitution setting out your committee structure, AGM requirements, process for elections and quorum restrictions? If you don't I think you should establish one.
Check this out communityimpactbucks.org.uk/wp-content/uploads/2018/03/CIB-Sample-Constitution.pdf

Thank you Trisher. There is a constitution but I haven’t seen it yet. At last years AGM the only people who turned up were some committee members , my daughter and myself. Subsequently we were elected onto the committee! The skiers are really good attenders but no one wants to run the club!

The reason I asked is because some constitutions make it a condition that you serve as vice chair for a year before you become chair and some say that if 2 people stand there must be a secret ballot at the AGM. It is amazing how many organisations have constitutions which no one ever looks at and which they are constantly infringing.
It's a good idea to offer something extra at the AGM, like show a film or photos taken during the year (people love to spot them selves).

Finding people willing at all to be on committees can be a huge problem. If they start making some members ineligible for committee membership there can be a huge risk of the club folding altogether. So many clubs just run on the good will of volunteers.
Constitutions can have been drawn up years ago and have little relevance for nowadays. Trying to update them is hard too. Who wants to go to meetings to discuss that?! I know it should be done but if the will's not there then clubs just have to get on as best they can.

Jane10 the OP is trying to find out if they can make one person chair in preference to another, I simply suggested that the constitution may offer a way of doing it.

If the head coach is a volunteer among volunteers I don't see what the problem is other than a personality clash. The constitution should steer the club but may not for reasons I outlined in my previous post.
Re committees- where there's a committee there's an an argument!