Upload & Sell: On
"Make them pay" based upon what? You have no contract with them. In order for there to be a contract, they would have had to make a payment along with signing.
That's incorrect. A signed agreement with a promise to pay is enforceable, particularly if you acted in reliance on that agreement to your detriment, as you seem to have done by turning away business for that date.
It's still unlikely to be worth attempting enforcement, but I'd also say people here do tend to fret too much about photographers enforcing their contractual rights. This client will never refer business to you, or even talk about you positively, no matter what you do. I give at least even odds that they could have paid you at signing, but wanted the option to talk to another photographer, you gave them the option, they took it and booked elsewhere, and then lied to you about the accident. There's likely no value in preserving a relationship with this couple, and the couple has cost you an opportunity.
Though I'd certainly investigate whether the groom was in fact injured first, and might not bother with an action if this claim were true, it would be legally appropriate for you to require payment of the retainer under the contract, and then to refund it if you re-book the date at full price. Whether the amount of your retainer makes the effort worthwhile is yet another matter, but I argue the risk to your reputation is a lot lower than people here worry it might be. It won't become a regional news story, much less a national one.