Help! How can we accommodate our guests after the ceremony if a restaurant can't accommodate us?

You don't "need" actual feed, but it would be very nice to have.

Since your event is mid week and likely not another event afterwards, why don't you ask the venue if they will allow you to remain longer and have food delivered.

If that doesn't work, does the inn you are staying out have a space that can accommodate you in this type of arrangement?

Unfortunately, this is likely a covid restriction so every restaurant is going to have the same seating limit for parties over 6.  If they don't abide by the restrictions they could get big fines and potentially lose their business license from the state.  I own a couple bakeries which have cafes in them and I can only have 50% capacity and no parties over 6 people right now which is an order from the state due to covid.

For now the champagne & cake may have to suffice.  Will the venue allow you any finger foods/appetizers with the cake?

Does anyone live in the area who can host something at their home?

Or you may need to have a dinner at a later date when restrictions are loosened bit.

The only other option would be to split up your group into 2 groups and if you do that you might not be seated together.  You also might not find a place who can do 2 groups at the same time based on capacity limitations.

Food at wedding receptions goes back to when people would have to travel by horse and carriage over treacherous terrain so they'd be sustained for the hard journey back home. Life isn't like that anymore and if you explain the situation surely your guests will understand that attending a wedding during a global pandemic comes with certain inconveniences. 

Why can they not accommodate 10 people?  Is this a pandemic thing? Have you told them it is a family group of ten from a wedding? 

Many places have a private room or section for a larger party. 

Book two tables. I would get two tables for six each. Give you and the groom seating at both tables so you can move back and forth.   lie if you have to, tell the waiter the other two will be here a little late  but we will order now. 

Will the hotel set up a room for you, have a caterer in with food. We did this at the  rehearsal dinner.

Make your reservation for a party of 6 and a party of 4. Or have food delivered to the venue.

Simple ...

Reserve 2 tables.

A table of 10 has zero ability for " togetherness" that 2 tables can provide.

Yes, you really should feed your guests as all that traveling needs more substance than a token snack at the venue.

Can you get away with two groups of five at the restaurant.  I don't have any good ideas other than separate the group.

This won't be popular,  but I don't care.  The pandemic thing is idiotic.  If someone is twitchy about attending, fine...they can just decline the invite.  

On your question, since this covers the lunch hour, etiquette says you should supply food.  It doesn't have to be elaborate or anything, especially for lunch, but I personally would serve something.  One thing you didn't make clear is whether the 10 restaurants turned you down because it was just cake and champagne.  In the middle of the week, even on short notice, there have to be places that can put something together for 10 people.  The only exception would be if you live in a very strict state that imposes this limit.  If so, are you near enough to a state line to try a  different state for the reception?  Also, what about the inn you're staying at?  Did you check with them?  You just need a small room and some very basic catering.  

Given you're getting married in lockdown, I doubt they'll necessarily be expecting a sit-down meal. 

If you want to do something, your best option is probably a picnic somewhere which you arrange yourselves (you could order hampers / deli collection locally for example)

Given that the indoor seating options are only 6, restaurants will have laid their seating out with this in mind, so won't be able to re-order for your party at the moment. 

you still have time to reschedule the whole wedding after the pandemic is over....