It is a particle. When you exclaim "Finally!"/"At last!", as an expression of relief (when something long awaited has happened) it is hardly ever used any other way. «Ну наконец-то» also works, «ну» being another particle usually used at the beginning of a sentence or as a filler word ("well").
When наконец is used as a part of a sentence, the -то is optional and used only for emphasis (so, you use it when not only something finally happened but you are very glad it did or it was high time it did).
Not obligatory particle which is emphasizing "наконец" https://en.wiktionary.org/wiki/-%D1%82%D0%BE#Particle
The meaning is a bit different. "Finally! Summer vacation!" - we've been waiting for weeks/months for summer vacation to arrive and it's finally here. Quite a normal thing to say. If you add the indefinite article, that implies that we haven't been having any summer vacation and this year, for the first time in five years, we're going to get one. Also a theoretically possible sentence but obviously not so frequently used.
Thanks a lot for such an interesting explanation! That's cool for me to know such nuances, thank you. Still my variant should be accepted.
Suppose you have been working for a long time in a standard, year round job and you take a job as a teacher so you will finally have "a summer vacation" (in the sense that school students use the phrase). Would you use the same expression or something different?
That would fall into the second case above, where you've not had a summer vacation for the last five years and now you're going to have one. So I think in this case it would be appropriate to use the article.
I think it would accurately describe the difference to say that if you use the article, it means that finally you have a summer vacation, while if you want to say that your summer vacation has finally arrived, you say it without the article.
Каникулы is a vacation for those who study (students), отпуск is a leave for working people. Also каникулы is used for deputies also meaning a leave.