I don't see why this wouldn't work, as nothing prevents you from taking a reaction on your turn.

Thus, the following sequence is valid:

1. The wizard uses their action to Ready the shocking grasp spell, which the familiar will deliver.
2. The familiar uses part of its movement to move within touch range of the target.
3. The familiar uses its reaction to deliver the wizard's shocking grasp spell. (The wizard also uses their reaction to use their readied action.)
4. The familiar uses its action to do something.
5. The familiar completes its movement.

The only catch here is that the wizard's reaction is consumed, and they have to maintain concentration until that time.

But yes, the familiar cannot use a readied move if the wizard wants to use the familiar to deliver a touch-range spell, as both would need to use the familiar's reaction, and the familiar can only take one reaction.

You must choose an action or movement, but you can omit other information.

Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it.

As we can see from this sentence, the minimum requirement is to choose an action or movement. I believe "the action" in this context is anything from Actions in Combat which includes:

one of the actions presented here, an action you gained from your class or a special feature, or an action that you improvise. Many monsters have action options of their own in their stat blocks.
When you describe an action not detailed elsewhere in the rules, the GM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.

So for instance "If X happens, I attack" and "If X happens, I use channel divinity" meet this requirement, whereas "If X happens, I do something" is not specific enough.

The PHB also gives us some examples from which we can make additional inferences:

1. If the cultist steps on the trapdoor, I'll pull the lever that opens it
2. If the goblin steps next to me, I move away

The first example is very specific.

In fact, there isn't any relevant information that we could add to "I'll pull the lever that opens it" to make the Ready action more specific: we can pull that one lever that opens the trapdoor mentioned in the trigger, and even if other levers were present we wouldn't be allowed to pull them.

The first example also shows us that we can choose an action implicitly, because in this example we chose the Use an Object action without naming it explicitly.

The second example is very vague.

There is plenty of relevant information that is missing from "I move away", such as whether we will walk, swim, or fly, and which path we will take. Therefore, the second example confirms that we can be vague and omit relevant information.

However, even this example is not completely devoid of relevant information, as "away" implies a general direction. This suggests that it's best to include at least some relevant information.

Moreover, eventually we must choose whether to walk, swim, or fly, and which path to take. So if our Ready action is vague we can make decisions during the reaction.

Your example.

You can ready an action for "When the Goblin moves, I Attack" as this meets the minimum requirement for the Ready action.
Because that Ready action omits relevant information, you must make decisions during your reaction, such as who to target with your attack(s) and which weapon to attack with.

In particular, you could choose to attack a Goblin, a Bugbear, or anyone else for that matter. However, you cannot unsheathe a weapon during your reaction because you can only use your free object interaction "during either your move or your action", which means that your choice is limited to the weapon(s) that you are wielding when the Goblin moves.