I'm afraid it's just a case of registering with as many supermarkets as will deliver in your area. And then just keep trying. I think Tesco release new slots at midnight ! Iceland are good and release slots here at 11am. Good luck - I know the feeling.
Oh...you could also look at deliveroo for same day...limited to Morrison's, M and S, and Co-op. Here in Plymouth anyway. You won't find a full range though.
I'm not classed as vulnerable and not over 70 but I live miles from any supermarkets and my car decided to throw a wobbly.
I have an important birthday party on Saturday
Keep checking. You never know of someone may drop out, but to be honest, you've probably left it too late. Can you get one of the slots when it's bagged and you pick it up?
It might be worth trying local independent traders, as many of them have introduced (or already had) delivery services. For example, our local corner shop accepts orders over the phone for delivery in a day or two's time, with only a £10 minimum order and just a £1 delivery charge.
Deliveroo will deliver food from M&S. Last week I had my delivery within 22 minutes. Fantastic service. Just download the Deliveroo App on to your mobile and type in M&S. You will have to register but it takes no time at all.