Many expiations are due upon me and I came to know that some restaurants undertake the distribution of the food of expiations. Can I give them money equal to the value of the expiations or should I distribute them by myself? Should the expiations be given all at once or at different times and over many days?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

If you are certain, or almost certain, that these restaurants distribute the food of expiations in the very way enjoined by Allaah The Almighty, then there is no objection to give them the required money so that they would distribute the food of expiations on your behalf to their due recipients.

But if you are uncertain, or almost uncertain, that they do so, it becomes due on you to distribute the food by yourself because certainty is an essential factor in clearing a person's liability.

As for the other part of your question, it is not a condition for the expiation to feed the needy persons all at the same time. What is due on you is to complete the number of those whom you should feed, even at different times. According to Al-‘Adawi if the person gives the food of the expiation to less than 60 needy persons, he should take back from each of them what is beyond the Mudd of foodstuff, if it is available to him, and complete the number (of recipients) to 60. [1:465]

Allaah Knows best.