What would be the correct word order to write it? I was thinking of
"milliōnēs diēs pluviī"
It gets a bit complicated, mostly for reasons that have to do with the way to express large numbers in Latin.
Word choice, "Rainy Days": diēs pluviī is indeed probably what you want for "rainy days." The phrase is used in the singular to describe days of rainy weather, for example, in Columella XI ("XI Kal. Febr. Fidicula vespere occidit, dies pluvius. ... V Kal. Febr. Auster aut Africus, hiemat, pluvius dies."). pluvius describes both things that are rain-bringing and things that are full of rain.
As a possible alternative, you might consider the adjective pluviosus, "rainy, full of rain," used in Pliny the Elder to describe a coming season that will be more rainy than usual ("ita nubilo occasu pluviosam hiemem denuntiat..."). Either adjective is fine; you should be able to use whichever sounds better to you without a loss of meaning.
Word choice, "One Million." Here you have a problem. As far as anyone can tell there is no simple word for "million" (i.e., one thousand thousands) in classical Latin (1). Modern European words for "million" are from the word milione, which first appears in medieval Italian. You have a couple of options here, depending on the sort of Latin you want to use in your translation:
in classical Latin you can only indicate a million by counting it out in thousands, usually as decies centena milia, i.e., "ten hundred thousands."
if you are comfortable with, or prefer, using late scholarly Latin then there is the late Latin noun milio, -onis which was coined by Renaissance scholars to express large numbers in scientific and diplomatic texts.
Putting the Phrase Together, Agreement in Gender: The noun dies is 5th declension, so the nominative singular and plural are both "dies;" you would decline the adjective pluvius, -a, -um to agree with it in case, number and gender. There is a complication here: in Latin "dies" is usually grammatically masculine, but in some contexts (especially when referring to set days or festivals) it can be grammatically feminine (see this prior question for more details). In most classical Latin texts it is used in the masculine, but sometimes it is used in the feminine when referring for example to holidays, calendar dates, or set dates and appointments. In most circumstances, you would probably use dies pluvii for the masculine form of dies; but if you are using it in a context where the feminine form is appropriate, you would use dies pluviae instead.
Putting the Phrase Together, Case: The other complication here has to do with the large number. Most numbers in Latin, when used to count the number of things (days, apples, horses, etc.), appear as indeclinable adjectives, which are relatively simple to use. Unfortunately large numbers often don't work this way. The usage for mille, milia is irregular and fairly complex (sometimes it works like the smaller numbers, sometimes it uses different rules and is used as a noun).
Without stepping through all of the rules, I'll venture that whether you choose to use the classical decies centena milia or the late Latin milio, your best bet is probably to use the number as a noun rather than an adjective, and to put dies pluvii/pluviae into the genitive case. (So you would say literally "ten hundred thousand of rainy days" or "one million of rainy days.") The genitive plural of dies is dierum; of pluvius is pluviorum (m.) or pluviarum (f.).
So, putting all that together, (1) you should choose whether you want to use classical "decies centena milia" or "milio"; then (2) choose based on context whether you want "dies" in the masculine (probably) or in the feminine (some contexts, e.g. set dates). Word order is flexible, and the adjective can be placed before or after dies. This could give you:
If you end up putting this into a larger Latin phrase or sentence, keep in mind that milia and milio here are declined nouns (here, in the nominative); so you may have to change their case according to the grammatical role that they play in the phrase or sentence. dierum pluviorum/pluviarum will remain in the genitive plural.