It depends on the type of vegetable, but generally most of them need at least 6-8 hours of sun a day. If you only get morning sun you're probably under 6 hours, which is a bit low for most garden plants.
But it's not all lost. There are vegetables that tolerate partial shade pretty well: lettuce, spinach, arugula, chard and cabbage do fairly well even with less sun. Salads and herbs like parsley and mint manage fine with 4-5 hours. Tomatoes, peppers and eggplants on the other hand are way more demanding and struggle with low light. Zucchini and cucumbers are somewhere in the middle: they'd prefer more sun but if you've got at least 5-6 hours they could still work.
My advice is to start with shade-tolerant crops and maybe arrange your garden so the sun moves across the area as much as possible during the day. If you get morning sun from the east, during warm season the light comes from other directions too. Anyway, vegetables that grow mainly for their leaves (lettuce, cabbage etc.) need less light than fruiting ones, so at least you know where to focus.