We are also in the invite queue for the Stripe Corporate Card, but decided to make a switch as we didn't know how long the wait would be. Brex is working on features for expense reporting and authorization is what were were told. While I don't think it's on par with Expensify, maybe one day in the future it will be which will mean one less service for us to manage.
For issuing cards, you can use Stripe nowadays. If you're doing this as part of your business (e.g. like Expensify is doing), its available to all US based businesses: https://stripe.com/issuing
The more convenient Stripe Corporate Card (which sounds like that may be your use case?) is still invite only: https://stripe.com/docs/corporate-card
(I myself am queued up for an invite on the Corporate Card, and using Chase Credit Cards in the meantime, which are okay, but not great)