If this is in the documentation, I missed it.

Is it possible to use arrays as item requirements for machine recipes? For instance, if I want the player to be able to use either minecraft:sugar OR harvestcraft:honeyitem as sweeteners in a food processing machine, can I just put an array like this?

    {
        "type": "item",
        "io-type": "input",
        "item": [

"minecraft:sugar",
"harvestcraft:honeyitem"
]
}

If not, is this a feature it would be possible to implement?

OK, but there's nothing in the wiki about how to use oredict. I've been through every page Ctrl+F'ing "dict", and got 0 results.

What's the syntax?

Should research Forge's OreDict, since that replaces one item with one that means many. You treat the oredict entry like it's the intended name instead.

Look at examples that are shipped with the mod.

Group them into an ore-dictionary-entry with Crafttweaker or related. That'd be the easiest approach. Atm this is not a thing due to - well - there being an alternative: OreDict.