I’ve sponsored a few sports teams and clubs, from school teams to the thrills of both league and non-league football.
At the level you are aiming at, it’s less convincing sponsors of the ‘advertising benefits’ and more on just getting local interest from local firms, who will be seen as supportive from their local audience. In addition, approach the big players in the town - Tesco etc.
Get prices together for the items to sponsor - kit, equipment, etc and seek sponsorship at cost, obviously with branding. One point to note, is contract for them for a year, always give the current sponsor the opportunity to renew for next year and factor in repairs/renewals into the quote.
I would also try and enlist the suppliers too, if not as sponsors, perhaps offering terms on payment - every little helps.
I can’t help on the grants available, but guess there will be some floating around, although imagine there will be few options these days.
Good luck!