Heat the olive oil in the Instant Pot using the saute function. When hot, add the chopped onion and celery. Saute, stirring often, until softened, 4-5 minutes. Add the garlic, press Cancel to turn off the saute function, and stir for 30 seconds.
Pour the vegetable broth into the pot and stir, scraping up any bits stuck to the bottom of the pot. Add all of the remaining ingredients: carrots, potatoes, green beans, corn, peas, diced tomatoes, Italian seasoning, paprika, salt, pepper and bay leaves. Stir.
Close the Instant Pot lid and turn the steam release valve to the sealing position. Set the cook time to 4 minutes at high pressure using the Pressure Cook or Manual function. The Instant Pot will take about 20 minutes to reach pressure and then the cook time will start counting down.
When the cook time ends, allow the pressure to release naturally for 5 minutes by just leaving the Instant Pot alone. Then, quick release the remaining pressure by carefully turning the steam release valve to the venting position using the handle of a long spoon. Once all of the steam has escaped and the pin drops down, carefully open the Instant Pot lid.
Remove the bay leaves. Stir the soup and serve.