Hy wil nou by die eiendomsagent aansoek doen.

Because wil is a modal verb meaning want to. In Afrikaans, when you use a modal verb, the other verb phrase usually goes to the end of the clause.

So:

• Hy wil ... doen = He wants to do ...
• aansoek doen is the verb expression meaning to apply / to make an application

That is why the sentence is arranged like this:

• Hy = he
• wil = wants to
• nou = now
• by die eiendomsagent = with / to the estate agent
• aansoek doen = apply

This end-position of the main verb phrase is very common after modal verbs in Afrikaans.

Aansoek doen is a fixed expression meaning to apply or to make an application.

Literally, it looks like:

• aansoek = application
• doen = do / make

So word-for-word it feels like do an application, but in natural English we translate it as apply.

Examples:

• Ek doen aansoek. = I am applying.
• Sy wil aansoek doen. = She wants to apply.

So you should learn aansoek doen as one unit of meaning.

Because aansoek doen is built from a noun + verb combination:

• aansoek = application
• doen = do / make

In Afrikaans, combinations like this often keep the noun before the verb. So aansoek doen is the normal order.

When another verb like wil is present, the whole expression goes to the end, but it still stays in that order:

• Hy wil aansoek doen.

Not:

• Hy wil doen aansoek.

In this sentence, by shows the person or place with whom the application is being made.

So:

• by die eiendomsagent = with / to the estate agent

Afrikaans often uses prepositions where English might simply use to or structure the sentence differently.

Compare:

• aansoek doen by iemand = apply with / to someone
• aansoek doen vir iets = apply for something

So the preposition depends on what you are talking about:

• Hy doen by die eiendomsagent aansoek vir die woonstel.
  = He is applying with/to the estate agent for the apartment.

Eiendomsagent is a compound noun, which is very common in Afrikaans.

It is made up of:

• eiendom = property
• agent = agent

Together:

• eiendomsagent = estate agent / real estate agent

The -s- in the middle links the parts. Afrikaans often joins words together into one compound where English might use two or more words.

Other examples:

• huisdeur = house door / front door
• tafeldoek = tablecloth

So eiendomsagent is perfectly normal Afrikaans spelling.

Yes. In Afrikaans, die is the definite article the for singular and plural nouns.

So:

• die man = the man
• die vrou = the woman
• die huis = the house
• die eiendomsagent = the estate agent
• die huise = the houses

Afrikaans does not have different words like English the versus gendered articles such as German der/die/das. That makes articles much simpler.

Nou is an adverb meaning now, and in a main clause it often comes after the finite verb.

So the pattern here is:

• Hy = subject
• wil = finite verb
• nou = adverb
• rest of sentence
• final verb phrase at the end

This is a very common Afrikaans word order pattern.

Compare:

• Hy werk nou. = He is working now.
• Hy wil nou werk. = He wants to work now.
• Hy wil nou by die eiendomsagent aansoek doen.

You may sometimes see adverbs in other positions for emphasis, but this placement is very natural.

Yes. Without the modal verb, it works as an ordinary verb expression.

For example:

• Hy doen aansoek by die eiendomsagent.
  = He is applying with/to the estate agent.

• Sy het aansoek gedoen.
  = She applied / has applied.

Notice how the form changes with tense:

• present: doen aansoek
• past/perfect: het aansoek gedoen

So in your sentence, wil is simply adding the idea of wanting.

In a subordinate clause, Afrikaans usually sends the finite verb to the end as well.

For example:

• Ek weet dat hy nou by die eiendomsagent aansoek wil doen.
  = I know that he now wants to apply to the estate agent.

Notice the difference:

Main clause:

• Hy wil nou by die eiendomsagent aansoek doen.

Subordinate clause:

• ... dat hy nou by die eiendomsagent aansoek wil doen.

So in a subordinate clause, wil moves toward the end too.

A rough English-friendly guide would be something like:

• eiendomsagent ≈ AY-en-doms-ah-ghent

A few notes:

• ei in Afrikaans often sounds a bit like ay
• g in Afrikaans is usually a throaty sound, not the English hard g
• agent in Afrikaans does not sound exactly like English agent

A more broken-down version:

• eien ≈ AY-en
• doms = doms
• agent ≈ ah-ghent

Pronunciation varies a little by speaker, but that rough guide will help.

It is fairly neutral and natural.

• Hy is the normal word for he
• wil nou is everyday language
• aansoek doen is standard Afrikaans
• eiendomsagent is the normal standard word for estate agent

So this sentence would fit normal spoken or written Afrikaans, depending on context. It is not especially slangy, and it is not unusually formal either.