[RPG] Can a lizardfolk monk use their bonus action to make an unarmed strike using their Bite trait

No. The Monk feature says that you can use the bonus action Unarmed strike after you use the Attack action with a Monk weapon or Unarmed strike. Shocking Grasp requires the Cast a Spell action, which does not qualify.

Strictly by Rules as Written and Intended, no.

By RAW, no. What you ruled doesn't work. Let's look at the important passages from the skills involved.

Martial Arts:

You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

Sneak Attack:

The Attack must use a Finesse or a ranged weapon.

While unarmed strikes count as weapon attacks, they are not weapons. It's an important distinction precisely because of situations like this.

Also note how Martial Arts doesn't say that it grants finesse. It could have and it would have been simpler. The fact they felt the need to write it the way that they did suggests that their intent was to keep the Monk Martial Arts skill separate from things like Sneak Attack.

By Rules as Fun, yes.

There's solid logic to your ruling and it gives one of your players an extra little oomph and, more importantly, fun. A ruling that goes against RAW isn't exactly wrong, especially in 5e where the rules talk a lot about the power of the DM.

What to do going forward.

What you should do in the future depends on how your table is run. If you want to stick to RAW or RAI, then you should probably tell the group at the next meeting that after further research you've decided that was a one time thing.

But if you don't mind breaking RAW and RAI a little because it's fun for the table, then there's nothing wrong with this interaction. Just keep in mind that you will be making your PC a little stronger than he's intended to be. Nothing wrong with that though, sometimes that's really what makes a campaign fun!