an someone drive a car without a license?
Well, every so often I see a story on a the news where a young child drove off in their parent's car.
But, you shouldn't be able to get ANY kind of job when you can't phrase a question properly. You already KNOW that you can get ANY kind of job you want -- all you have to do is convince a hiring manager to give you the job. Maybe the guy is your neighbor's brother, so he decides to try you out.
But, yes, even without connections, it's within the realm of possibility. Getting ANY kind of degree, if you get good grades and it's from a good school, is an accomplishment. The idea would be to find a web developer position that is somehow related to psychology.
So, do psychologists need websites?
But, keep in mind that you'll have competition. So, why should they hire you rather than someone with a CS degree. If you manage to get an interview, you'll have to be able to answer that question.
All of this stuff I said is pretty obvious.
By the way, a portfolio show examples of work that you did FOR PAY. If you've never gotten paid to develop a website, you are NOT going to have a portfolio. They don't expect an entry-level candidate to have one. They will base their decision on your school record and your ability to sell yourself.