Are there any sources that suggest that there is only a set amount/number of souls which can be born into bodies before the arrival of Moshiach or before Olam Haba?
If yes, can Moshiach even come early if there are souls yet to be born? What would happen to souls that did not experience Olam Hazeh?
Yes. Yevamot 63b (with Steinsaltz translation and interlinear commentary:
אָמַר רַבִּי אַסִּי: אֵין בֶּן דָּוִד בָּא עַד שֶׁיִּכְלוּ כׇּל הַנְּשָׁמוֹת שֶׁבַּגּוּף, שֶׁנֶּאֱמַר: ״כִּי רוּחַ מִלְּפָנַי יַעֲטוֹף וּנְשָׁמוֹת אֲנִי עָשִׂיתִי״. תַּנְיָא, רַבִּי אֱלִיעֶזֶר אוֹמֵר: כׇּל מִי שֶׁאֵין עוֹסֵק בִּפְרִיָּה וּרְבִיָּה — כְּאִילּוּ שׁוֹפֵךְ דָּמִים, שֶׁנֶּאֱמַר: ״שׁוֹפֵךְ דַּם הָאָדָם בָּאָדָם דָּמוֹ יִשָּׁפֵךְ״, וּכְתִיב בָּתְרֵיהּ: ״וְאַתֶּם פְּרוּ וּרְבוּ״.
Rabbi Asi said: The Messiah, son of David, will not come until all the souls of the body have been finished, i.e., until all souls that are destined to inhabit physical bodies will do so. As it is stated: “For the spirit that enwraps itself is from Me, and the souls that I have made” (Isaiah 57:16). It is taught in a baraita that Rabbi Eliezer says: Anyone who does not engage in the mitzva to be fruitful and multiply is considered as though he sheds blood, as it is stated: “Whoever sheds the blood of man, by man shall his blood be shed” (Genesis 9:6), and it is written immediately afterward: “And you, be fruitful and multiply” (Genesis 9:7).
The Moshiach can't come "early", but he can come "earlier". For example, this gemara shows that people have the option to delay Moshiach by refusing to have babies.
