Påmeldingen stenger med en gang innleveringen begynner.

Because they are in the definite singular form.

In Norwegian, instead of usually putting a separate word like the before a noun, you often add the definiteness to the end of the noun:

• påmelding = registration / sign-up

• påmeldingen = the registration

• innlevering = submission / hand-in
• innleveringen = the submission

So this sentence is talking about a specific registration period and a specific submission period, not registration and submission in general.

Påmelding usually means registration, sign-up, or enrolment.

In this sentence, påmeldingen most naturally means the registration period or the sign-up process. So Påmeldingen stenger means something like:

• Registration closes
• The sign-up closes

Even though stenger literally means closes, in English we often phrase it more naturally as registration closes.

Yes, stenge can mean to close or to shut.

It can be used:

• transitively: closing something
  
  • Hun stenger døra. = She closes the door.
• intransitively: something closes
  
  • Butikken stenger klokka åtte. = The shop closes at eight.

In your sentence, it is used in the intransitive sense:

• Påmeldingen stenger = Registration closes

This is very normal Norwegian.

Here, med en gang means as soon as or immediately when.

Literally, it can look like with one time, but that is not how you should understand it. It is an idiomatic expression.

So:

• med en gang innleveringen begynner = as soon as the submission begins

Depending on context, med en gang can also mean immediately in other sentences:

• Kom hit med en gang! = Come here immediately!

Because Norwegian often uses the present tense to talk about scheduled or definite future events, just like English sometimes does.

So:

• Påmeldingen stenger ... = Registration closes ...
• innleveringen begynner = the submission begins

This can refer to a future event if the timing is understood from context.

Norwegian does this very naturally with timetables, rules, and planned events.

Because the sentence is talking about the submission period/event itself, not the action to submit in a general sense.

Compare:

• innlevering = submission, hand-in, the act/process of handing something in
• å levere inn = to hand in / to submit

So:

• innleveringen begynner = the submission period begins
• å levere inn begynner would not work here

Norwegian often uses nouns like påmelding, innlevering, undervisning, betaling, etc. where English might sometimes prefer a verb or a different structure.

It depends on context.

Innlevering can refer to:

• the act of submitting something
• the submission itself
• the submission/hand-in period or deadline window

In this sentence, because it says begynner, it most likely refers to a submission period or hand-in phase, not a single submitted item.

So the most natural sense is probably:

• the submission period begins
• the hand-in opens/begins

Because med en gang already introduces that time relationship.

In English, you often need as soon as or when, but in Norwegian med en gang can do that job by itself:

• med en gang innleveringen begynner = as soon as the submission begins

So you do not need an extra når here.

The structure is:

• Påmeldingen = subject
• stenger = verb
• med en gang innleveringen begynner = time clause / adverbial phrase

Broken down:

• Påmeldingen stenger = Registration closes
• med en gang innleveringen begynner = as soon as the submission begins

So the full sentence says that one event happens immediately when the other event starts.

Yes, often you could.

For example:

• Påmeldingen stenger så snart innleveringen begynner.

This would also mean Registration closes as soon as the submission begins.

A small nuance:

• med en gang often feels like immediately when
• så snart often feels like as soon as

In many contexts, they are very close in meaning.

Yes, it is standard Bokmål, and it is grammatically natural.

It sounds like an administrative or academic sentence, the kind of thing you might see in:

• course information
• exam rules
• event registration instructions
• online portals

A very natural English-style paraphrase would be:

• Registration closes as soon as submission begins.
• The sign-up closes once the submission period starts.

So yes, it is perfectly normal Norwegian.