Sit down meal

I would find that unusual as a guest. If a good friend told me about the situation I wouldn't mind, but they'd either have to be a good friend or a close relative. Could you have a small buffet instead of a sit down meal?

I would look at alternative options like a buffet or food truck so you don’t have to have a sit down meal if budget is that tight. It isn’t the done thing to ask people to pay I am afraid as they are your guests.

I’ve been to a wedding before where the service was registry office and the meal was at a local restaurant. Due to unexpected circumstances nearer the wedding, The bride and groom asked guests to pay for their meal and not to give wedding gifts.
I have also heard of couples asking guests to donate a dish for a buffet style type meal.
I think it all depends on how many people you are having etc.Another option could be to have a smaller reception so you do not have any many people to feed?

I have been at weddings where there was no food served but there was a free bar for a bit of time. That might work out cheaper

or like others have said, do a buffet

I've been to a couple of weddings where the couple served nibbles or cake immediately after the reception for everyone who came and then had a small private dinner with immediate family later in the day. That might be something to consider.

Alternatively, book somewhere informal like a village hall and ask people to do a 'bring and share' lunch.

I think either of these options would be better than asking people to pay for a meal, especially as they have likely already gone to some expense to attend your wedding in the first place.

I think this would come across pretty rude tbh... I know weddings are EXPENSIVE but remember it's your day and you're the one that gets to choose the scope. It's actually kinda expensive to be a guest (accommodation, taxis, outfits, gifts, possibly annual leave) and as much as guests want to be their to celebrate with you, it's a tough sell to ask them to also shell out for a meal that they'll likely have very little say in.

As others have suggested, buffets can be cheaper. Another option is to cut your guest list down, this is the easiest way to save money - invite less people so that you know your budget will stretch to be able to feed them.

My cousin got married at a registry office and then went to a local beefeater for the meal and everyone paid for their own. No dramas at all she did not have any money for her wedding and everyone understood