There is nothing built in to Store that would allow you to prevent a particular card type from being processed at the module level.
If the gateway required the card type you would be able to use a select and omit any card types you did not want to support. So for example you could do
<option value="">Select one...</option>
if you only wanted to support Visa, MasterCard and Discover but not Amex. Stripe however does not require the card type to be submitted for a successful transaction. I also don't think that submitting the wrong card type would cause the transaction to fail as a result (even if you added
data-stripe="card_type"). Having the select however would give a visual cue to the end user that Amex is not accepted.
If a JS option is out of the question you should be able to use the store_payment_request_start hook to do any additional processing/validation before the transaction to sent to the payment processor.