Is it possible to omit 'to' from the following sentence, 'This letter was sent to him yesterday.'?

This letter was sent him yesterday.

This is the passive. The passive form does not take an indirect object; it takes adverbial prepositional phrases as complements.

In full, this would be

This letter was sent (i)by me (ii) to him (iii)yesterday.

In “I sent him this letter yesterday”, the active form does take

(i) two objects, (a direct and an indirect) or

(ii) an object and an adverbial prepositional phrase as a complement: I sent this letter to him yesterday”.

(This is known as the "dative shift" - the theory being that in the example sentence in the active, "him" is in the dative grammatical case and implies to [or for] him.)

Having said that, “This letter was sent him yesterday” can be heard from some native speakers, but I would not expect it to appear in business or formal written or spoken English.

Syntactically, OP's text is fine with or without to, but I certainly find this NGram usage chart interesting...

I'm fairly sure that as a general principle, the prevalence (and semantic significance) of prepositions in English has been increasing over the centuries. The specific verb in our sights here is to send, but it's worth noting this...

If a language has only one ditransitive trivalent verb (on the basis of any feature of formal transitivity), then that verb is ‘give’ _{Kittilä, (2006: 604)}

(Exactly the same usage shift is seen with was given me by, now largely supplanted by was given to me by.)

It is grammatically possible but semantically preposterous, except in some dialects.†

To see this more clearly, let’s look your ditransitive verb sentence in the active and passive voices. For illustration purposes, I’m changing him to Kim and adding the agent Pat to your passive voice version:

Here it is with the direct object following the verb and the second object in a to-prepositional phrase:

Passive: This letter was sent to Kim [by Pat].

Active: Pat sent this letter to Kim.

All good. Now let’s look at this with an indirect object in front of the direct object:

Passive: Kim was sent this letter [by Pat].

Active: Pat sent Kim this letter.

All good. Now let’s omit to, as in your second example:

Passive: ? This letter was sent Kim [by Pat].

Active: ?Pat sent this letter Kim.

Not so good. Here we have the absurd situation of the letter receiving a package — which contains Kim — from Pat.

_{† See:
Give it me!: Pronominal ditransitives in English dialects
Ditransitive clauses in English with special reference to Lancashire dialect}

_{Further reading:
Lies Your Grammar Teacher Told You: Indirect Objects}