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One of the entries on SpanishDict.com for the verb entrar is this:
to get: Me está entrando sueño; me voy a echar una siesta. I'm getting drowsy; I am going to take a nap.
Can someone explain why the third-person of estar is being used here? I was under the impression that estar used as a pronominal verb means "remain," not "become." Wouldn't it be better to use ponerse or convertirse en anyway?
The dictionary you consulted is giving an idiomatic use of the verb "entrar." Yes, ponerse (I think you had a tiny typo with poderse) or convertirse can indeed be used for transitions, such as
Ya empiezo a ponerme más relajado/a.
La tormenta se está conviertiendo en algo serio.
But it is generally the case that an idiom could be substituted with a more straightforward word or phrase.
Remember, an idiom is something that does not take its meaning from the direct sum of the parts. Instead, the special grouping of words takes on a special meaning that's not obvious from putting the bits and pieces together.
As you gain experience with Spanish idioms, you will find that they give a special kind of pleasure.
To answer the part of your question that asks about estar: yes, it is possible to say
Me entra sueño
(without "estar" being part of the sentence).
The reason for "estar" being part of the sentence is that it is being used to construct a present progressive or present continuous tense. Compare
a) He gets sleepy every day around 4 pm.
b) He is getting sleepy now -- he woke up very early this morning.
In this case the pronoun does not belong to the verb estar but rather to the verb entrar. Consider the following rephrasing of the sentence (it sounds really weird, but it is still correct):
El sueño está entrándome (a mí).
(If you want, for the sake of the example, you can translate this sentece word by word as follows: Sleep is attacking me. Note: attacking is not a good translation for "entrar", but it matches the grammar of the spanish sentence).
As you can see more clearly in the example above, el sueño is the subject, thus the 3rd singular form of estar (está). Besides, the subject el sueño is performing an action (in this case entrar), and in this case that action entrar has an indirect object which is me, thus the pronoum me (or a mí).
The verb entrar with this meaning is explained by the entries 3 and 5 of the DLE, although in your example is not used literally but figuratively:
3-. intr. Dicho de una cosa: Encajar o poderse meter en otra, o dentro de otra.
5-. intr. Penetrar o introducirse.
First, take está entrando as a verb phrase: it's the way of expressing the progressive aspect of entrar (progressive = ongoing action). The core meaning is that of entrar.
Second, forget the literal translation; you have to think of the phrase as a whole. Me está entrando sueño means I'm getting sleepy, but neither the words nor the grammar will match.
The grammar here is parallel to that used for gustar and also for parecer and some other verbs that show an involuntary state or change of state. In Spanish the normal word order for these verbs is different from that of most phrases: indirect object, verb, subject or complement (e.g.: me entra sueño, nos gusta bailar, le parece incorrecto).
If you were to rearrange the phrase using the most common word order in Spanish, it would be Sueño me está entrando, which is ungrammatical because a bare noun cannot be used in this way. In English you could however say Sleep is getting into me, which is odd but readily understandable.
If you wanted to mimic the English phrase using the equivalence get = ponerse, you could also say Me estoy poniendo somnoliento, but that's an absolutely weird phrase, even if it's grammatically correct, and somnoliento is a formal word very unlike English sleepy.
Me está dando sueño is a common synonym. By itself it means the same as Me está entrando sueño, and the verb will always agree with sueño (i.e. it will be third person singular). It can also be used with other syntax, where sueño is not the subject but the direct object, and dar works as expected: Los medicamentos me están dando sueño.
There is no way you can use convertirse, in any conceivable phrase, to express the meaning of "I'm getting sleepy".
This is because "Me esta entrando sueño" is not the literal translation of "I'm getting drowsy".
The literal translation is "Me estoy poniendo adormecido/me estoy adormeciendo", and while the translation is correct, it is not used a lot.
Also the literal English translation of "Me esta entrando sueño" is "the sleepiness is getting into me" (not very idiomatic).
You can say that what you are getting is the translation of the idea but not the translation of the words.
Also the verb "Estar" is only one of the two forms of the verb "be":
Ser/Estar
