Is it theoretically possible that electrons are made up of quarks just like protons and neutrons?

Yes! The electron definitely doesn't have to be fundamental. In fact the LHC does searches that rule out electron compositeness up to a certain energy scale.

If you're trying to make up the electron out of Standard Model (SM) quarks, you are going to run into problems:

  1. Why is the electron being bound together at such a higher scale than the typical strong force (or QCD) confinement scale? This suggests that the force holding the electron together is an exotic force. That means whatever quarks are living inside the electron need to be charged under this exotic force (we are now building a BSM model).

  2. If the exotic force confined to form the electron, when the exotic force confined, how do we know we didn't trigger QCD breaking? Worse, in your example, how do you know we didn't trigger electroweak symmetry breaking (EWSB) a la technicolor models? The electron compositeness scale has been ruled out up to scales far above the EWSB scale.

I'm not 100% sure you can't find a clever way to address these two points, but it's hard for me to see a fruitful model that manages to get around these constraints.

The easier way to build a model of a composite electron is to do it with truly exotic fermions that aren't charged under the SM QCD group. You can think of dark quarks charged only under a dark QCD that bind to form the electron.

Another question for further reading: how is the electron so light if its compositeness scale is so high? What happened to the binding energy? Baryons tend to be living at the scale of QCD confinement in the SM.