I know that electrons' only interaction with the higgs field is due to ambidextrousness and the weak hypercharge which is provided by the higgs boson, but how do quarks (specifically up and down) interact if they are confined by gluons and therefore have a fixed chirality? How do they have mass to?
This blog post is a good one to get a feeling of the Higgs mechanism.
One should definitely separate the Higgs field from the Higgs boson. The Higgs boson is an elementary particle , attendant to the existence of the Higgs field as its excitation, and acquires its mass as all the other particles in the standard model table.
The Higgs field that gives the elementary particle masses in contrast to all the other fields posited by field theory, which are zero everywhere unless a "particle" exists as an excitation of the field, is nonzero everywhere.
Please read the link to understand that the masses are not acquired by interaction of the type "Higgs+electron" or "Higgs + quark" Feynman vertices, but by the change in the basic vacuum expectation value through which the other fields, photon, electron etc , exist and move.
So chirality has nothing to do with the acquisition of mass from the quarks and the other fermions through the Higgs mechanism. (In any case chirality is definite only for zero mass fermions , whether quarks or leptons.)
I'm pretty sure the simple answer is that they (quarks) do get their intrinsic mass via interactions with the Higgs field (via exchange of "Weak Hypercharge").
However, given that quarks cannot exist on their own, composite particles made of quarks (protons, neutrons, etc.) will end up getting far more of their intrinsic mass from the binding energy of the gluon field (i.e. particles made of quarks end up getting most of their mass via totally separate mechanism than the Higgs; via the exchange of gluons and virtual particles).
Try reading up on QED (Quantum chromodynamics) and/ or check out this related video from Veritasium if your interested in learning more on this subject: