Why is it that is some places in Melachim Eliyahu is spelled with a Vav and sometimes the Vav is missing? I have looked in as many sources as I can think of and can not find an answer.
Examples: Melachim Aleph Perek 17, 18,19,21,: with Vav Melachim Bet Perek 1: no Vav Melachim Bet Perek 2, 3, 9, 10: with Vav
As mentioned here, different spellings of names that refer to the same person are not uncommon in Tanach. Deeper, esoteric meanings are associated with the changing of spellings. As a general rule, therefore, you will not find commentators of the p'shat approach that will address these spelling changes.
In this case , the most famous answer is that of the midrash quoted by Rashi:
בחמשה מקומות נכתב מלא ואליהו חסר בחמשה מקומות יעקב נטל אות משמו של אליהו ערבון שיבוא ויבשר גאולת בניו
In five places ["Yaakov"] is written in full form [with a vav] and "Eliyahu" is written in lacking form [without a vav] in five places. Yaakov took a letter from Eliyahu's name as a guarantee that he will come and announce the redemption of his children.
For a more extensive explanation of this midrash, see here
