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Recently, in the first session of a new campaign, my level 1 wizard found himself held captive with his hands and feet tied by rope. The DM made it clear that I could not cast spells/cantrips that required a Somatic component.

However, my familiar was already summoned and was temporarily dismissed using the action specified in the find familiar spell (PHB pg. 240). I wanted to have my familiar reappear within 30 ft. of me as per the reappear action specified in this spell (I wanted it to help me get out of my binds). The DM told me that since the ritual/spell required somatic components, and even though I wasn't casting it again, if I wanted my familiar to reappear I would need to be able to make a somatic movement (thus I could not make my familiar reappear and help).

Does using the reappear action in find familiar require the use of my hands, or is it purely telepathic?

One of my primary reasons for picking find familiar as a spell was its ability to help escapes by reappearing and disappearing, which is a lot less doable if it requires somatic movement.

Find Familiar

As an action while it is temporarily dismissed, you can cause it to reappear in any unoccupied space within 30 feet of you.

This is not casting the spell and so any restriction on Components you can currently use is not relevant.

All this says is that causing your familiar to reappear take your action for the turn in which you do it. As written, it does not require you to do anything in particular so if you are able to take your action (i.e. you are not Incapacitated) you should be able to do this.

However, if your DM wants to impose limits on this beyond what is in the spell description, that is their prerogative.

Per the Find Familiar spell:

As an action, you can temporarily dismiss your familiar. It disappears into a pocket dimension where it awaits your summons. Alternatively, you can dismiss it forever. As an action while it is temporarily dismissed, you can cause it to reappear in any unoccupied space within 30 feet of you.

There is no mention of any component required; verbal, somatic, or material. The accepted school of thought is that spells do exactly what they say; nothing more, nothing less.

The same "as an action" wording occurs in the same spell:

Additionally, as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses.

Once again, no mention of anything other than using an Action.

As a counterpoint to the DM, the casting of the spell requires an hour of time and 10gp worth of incense. Since both of these features happen in the span of an action, which is generally one round so 6 seconds, there is no correlation between the initial casting of the spell and using the features/benefits from its casting.

Unfortunately, there are no spell descriptions that specifically say that something more is required beyond "as an action" so this cannot be proven via example. In other words, no spell says, "As an action, point your finger at a new target."

The rules do not specify the nature of the action required so it is up to the DM

We know that certain interactions are considered free. There are things like opening doors, removing a ring from your finger, withdrawing a weapon from your backpack, etc (these and other free interactions are listed in the box on page 190 of the PHB).

The fact that summoning and dismissing the familiar requires an action suggests that there is something about it that makes it more complex than these more minor interactions.

Neither the general rules nor the rules specific to the Find Familiar spell explicitly state that resummoning the familiar require any VSM components but the DM is empowered to interpret or arbitrate the rules as s/he sees fit (DMG page 5).

It is also with pointing out that, being bound, you may be subject to the Incapacitated condition which prevents the taking of actions or reactions. Or you could simply be Restrained, which does permit the taking of actions.

While I agree with the points made by other posters, it is up to DM discretion.
Would you be able to summon/conjure/reappear your familiar from the pocket dimension back into the Material Plane if you were in an anti-magic field?

I don't think so, there is some magic involved.

But since it doesn't say it requires components, I'm on your side.

What the DM should've done is let your captors also gag and blindfold you.

Because you do need to be able to see the spot where you want to resummon your familiar and seriously who doesn't gag his wizard prisoner?