Scrabble's standard for what constitutes a word, as Janus Bahs Jacquet notes in a comment, is whether or not it appears in a dictionary. For organized play, specifically, the word must appear in an official dictionary currently published by Merriam-Webster; various other dictionaries have been used in the past. This is rather different from a writer's definition, or a grammarian or other linguist's definition of what constitutes a word.
No dictionary can contain every possible variation of every possible valid word. Modern dictionaries do attempt to include words and meanings as they are commonly used; there is a long editorial process of identifying new words, finding evidence for their meanings, and choosing whether or not to include them. But while ailer is superficially analogous to, say, wailer, it is not in common use. I did not turn up a single example in COCA or the BNC, and the only examples I found in Google Books after 1900 are names or transcription errors. Given publishing constraints, a word that may be understood in context, but is rarely ever encountered, may be excluded from dictionaries uncontroversially.
Lastly, as John Lawler alludes, the fact that jumper or maker exists does not require that ailer also be accepted universally. One may be a sufferer, but not really a convalescer. Or someone who was ailing and is recovering may be a survivor, which brings up the point that the same sound may be represented differently in the written word; see What’s the rule for adding “-er” vs. “-or” when nouning a verb?, Pedlar vs. peddler and other questions tagged as agent-noun-suffix. Perhaps the Official Scrabble Players Dictionary does list it— but as ailor. [Rhetorical point. It doesn't.]