Learn Hardcore Finnish: Automaatti ei anna vuoronumeroa, joten pyydän virkailijalta apua. - The machine doesn’t give a queue number, so I ask the clerk for help.

Because in Finnish, the object is often in the partitive when the action is incomplete, ongoing, non-resultative, or—very commonly—when the clause is negative.
Here the clause is negative (ei anna = does not give), so the object becomes partitive: ei anna vuoronumeroa.

Compare:

Automaatti antaa vuoronumeron. = The machine gives (you) a queue number. (a completed, successful result)
Automaatti ei anna vuoronumeroa. = The machine does not give (you) a queue number. (negative → partitive)

How does the Finnish negative work in Automaatti ei anna?

Finnish forms negation with a special negative auxiliary verb that conjugates, plus the main verb in a special form.

ei = negative verb, 3rd person singular (he/she/it)
anna = main verb antaa in the connegative form (it doesn’t show person)

So:

minä en anna = I don’t give
sinä et anna = you don’t give
hän ei anna / automaatti ei anna = he/she/it doesn’t give

Why is there a comma before joten?

Because joten (so / therefore) introduces a new clause, and Finnish normally separates two full clauses with a comma.

Structure:

Automaatti ei anna vuoronumeroa, (clause 1)
joten pyydän virkailijalta apua. (clause 2)

What exactly does joten mean, and can it be replaced?

joten means so / therefore / for that reason, connecting cause → result.

Common alternatives (with slightly different style/structure):

Automaatti ei anna vuoronumeroa, joten... = …so…
Automaatti ei anna vuoronumeroa, niin... = …so… (more spoken)
Koska automaatti ei anna vuoronumeroa, pyydän... = Because…, I ask… (cause-clause first)

Why is pyydän in the present tense—does it mean “I will ask” or “I ask”?

Finnish present tense often covers both present and near-future / intended action, depending on context.
Here it’s a natural “real-time decision” meaning: so I ask / so I’m going to ask.

pyydän = I ask / I request (right now / as a plan in this situation)

If you wanted to emphasize “I’m going to,” you could also use expressions like aion pyytää (I intend to ask), but it’s not necessary here.

Why is it virkailijalta? What case is that, and what does it imply?

virkailijalta is the ablative case (-lta/-ltä), which often means from someone/something (especially a person as a “source”).

pyytää joltakulta jotakin = to ask/request something from someone
So:
pyydän virkailijalta apua = I ask the clerk for help (literally: request help from the clerk)

Other “from” cases exist (like -sta/-stä), but -lta/-ltä is the normal choice with pyytää in this pattern.

Why is apua in the partitive case?

apu (help) is typically used as a mass/uncountable noun in Finnish, and it commonly appears in the partitive when you mean “some help” in a general sense.

Also, many common request phrases take partitive:

pyytää apua = ask for help
tarvita apua = need help
antaa apua = give help

You can see avun (genitive/accusative) in some contexts, but apua is the default for “help” in everyday Finnish.

The sentence has no word for “me” (like “to me”). How do we know who is supposed to get the queue number?

Finnish often leaves this kind of recipient implicit when it’s obvious from context. In a ticket-machine situation, “doesn’t give a queue number” naturally implies “to me/us”.

If you wanted to make it explicit, you could add an allative:

Automaatti ei anna minulle vuoronumeroa. = The machine doesn’t give me a queue number.

But it’s very common to omit it.

Can I change the word order, like Pyydän apua virkailijalta?

Yes. Finnish word order is flexible, and changes usually affect focus/emphasis rather than basic meaning.

pyydän virkailijalta apua = neutral: from the clerk, help
pyydän apua virkailijalta = slightly more emphasis on apua (help)
virkailijalta pyydän apua = stronger emphasis/contrast: it’s specifically from the clerk that I ask

What’s the difference between vuoronumero and vuoronumeroa/vuoronumeron in general?

They’re different cases of the same noun:

vuoronumero (nominative) = the basic dictionary form; often subject or “label form”
vuoronumeron (genitive / accusative) = often a total object (“the whole thing”, successful result): antaa vuoronumeron
vuoronumeroa (partitive) = often partial/ongoing, or used in negative clauses: ei anna vuoronumeroa

So the choice is mostly about grammar and meaning nuances (especially negativity and “complete vs incomplete result”).