Since the modem/GPS card on the Vita 1100 series uses a standard miniPCIExpress connector (USB/SIMcard pins only), albeit with a non-standard ¾ length card, it can be replaced with another USB-based device. And while the official firmware is rigged to only support USB mass storage for PSTV recovery, that's nothing a 3rd party driver can't fix, right? :)
The PSVSD, invented by Yifan Lu in August 2017, is an ordinary microSD to USB card reader designed for this form factor. It can optionally be combined with a flat cable microSD extension (or even a reverse SD to microSD adapter) to allow external access to the card via the SIM slot.
Other mass storage devices are also viable...
On the Vita 1000
The non-3G OLED system lacks a miniPCIExpress connector and a few discrete inductive filters (replaceable with a short) completing the power and data lines, however it is inconclusive whether this is sufficient to restore USB support.
VitaShell offers an USB storage mounting feature in its main menu, originally invented for PSTV users and in fact pre-dating the PSVSD concept.
The original driver by Yifan Lu.
Main article: Vita:SD2Vita
The newest and most flexible storage driver.