Can you submit a batch of transactions using eAuthorize?

This is a yes and no type answer. eAuthorize does not support a batch upload of a single file containing multiple transaction requests for the gateway to process. Receiving reporting back on the success or failure of the transaction is part of the problem when doing a batch upload. There is no way built into the plugin to make this possible.
However, if you have a group of transactions in your database that you wish to run, you could consider using a looping script in FileMaker to execute the transactions one after another. This way, each transaction would be handled individually by the plugin, the results would be brought back into your FM database, and the “batch” would be processed after the loop completed going through each transaction.
We do know of a number of clients that are using such a looping method with success. It also has the advantage of allowing the logic in your database to execute for each transaction based on the result. If the transaction is successful, you can proceed with order fulfillment and so forth. If the transaction is denied, you can flag the account for follow up.