Update: Since around January 2018, it is now possible to search using the email parameter on Stripe. See the accepted answer.
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Stripe API does not supports any search-by-email feature. They have this search in their dashboard but not released any to API; from the architectural concept it seems that there is no possibility or plan from stripe to include this in API; every object in their API is retrievable only by that specific objects id given by stripe while its created. May be, they have kept it as a scope for third party application developers involvement!!
So, the obvious solution is to store the customers in your own database that you want to be searchable in future - as Simeon Visser has said above
btw, for a workaround, if you already have used the stripe API a lot and there are many customer data which you now need to be searchable - the only way is to go thru the 'List all customers' functionality of API & build the database for your own purpose; ofcourse, you've to use pagination shortcut to iterate thru the whole list for doing so.
$customers = \Stripe\Customer::all(array("limit" => 3));
foreach ($customers->autoPagingIterator() as $customer) {
// Do something with $customer
}