Without a proprietary Sony memory card, you cannot download digital games, install patches, or even save progress in many physical games.
Whether you can download games on a PS Vita without a memory card depends entirely on which model you own and whether you are willing to use custom firmware. Out of the box, the answer is often "no" or "very limited," but modern software modifications have changed the landscape for this classic handheld. PS Vita 1000 (The OLED "Fat" Model) can you download games on ps vita without memory card
Using modern exploits like HENlo, you can now "unlock" a hidden partition in the Vita 1000's internal memory. This creates a 1GB internal memory card partition that mimics the storage found in the newer models, allowing you to install small games or basic homebrew without any physical card. PS Vita 2000 (The "Slim" Model) Without a proprietary Sony memory card, you cannot