What ways are there for a wizard in D&D 5e to use a shield?
The only option I could think of was multiclassing which does not seem worth it.
The only ways for a wizard to gain shield proficiency are either multiclassing, or the Moderately Armored feat. Since Moderately Armored requires you to be proficient in light armor, most wizards will need either a multiclass or another feat (Lightly Armored).
Note that you can use a shield without proficiency - you just won't be able to cast spells, and you'll have disadvantage on any attack roll, saving throw, or ability check that involves Strength or Dexterity.