Your understanding is correct. Note that your understanding is not a translation of the source text.
Your questions
What is the subject of the above sentence?
"Subject" implies the subject of the verb. For the main verb on the end, なります, the grammatical subject matches the topic of the sentence: インターネットの手続き, or "the procedure for getting an internet connection".
Is で in お客様のほうで identical to で in 手で食べます?
Broadly speaking, yes. In お客様のほうで...とって頂く, the toru-ing is being done by the side of the guest, to torture the English grammar a bit. In 手で食べます, the taberu-ing is being done by hand.
How to decompose the sentence into smaller grammatical parts?
For that, read the next section.
Full analysis
Let's break it down.
Taking it apart
インターネットの手{て}続{つづ}きはお客{きゃく}様{さま}のほうで取{と}って頂{いただ}く形{かたち}になります。
First off, what are we talking about? What is the topic of this sentence? Well, in simplified terms, we look for the は:
インターネットの手{て}続{つづ}きは
So we know we're talking about the the internet (インターネット) procedure (手続き), or in this context, the procedure for getting an internet connection.
Now let's look at the rest of the sentence:
お客{きゃく}様{さま}のほうで取{と}って頂{いただ}く形{かたち}になります。
This gets a bit more complicated.
First off, we have お客{きゃく}様{さま}, a quite polite way of saying "guest". The お and 様{さま} are social register (kind of like politeness) cues that tell us that this whole sentence is probably spoken by someone in a host or service position, talking to or about the guests.
The のほうで here tells us that whatever is happening is something that happens literally "on the side of" the guest, or "on the part of" the guest -- or more idiomatically, something that the guest takes care of.
とって頂く breaks down to とって or "take", here more like "take care of" or "handle", plus 頂{いただ}く, humble for "receive", here used to mean that the speaker is getting someone to do something for them. So the speaker is getting someone else to do the "taking [care of]".
形{かたち} is literally "shape" or "form". The になります on the end literally means "becomes [whatever came before]". As a set phrase, 形{かたち}になります means "this is the way things become", and this is often used idiomatically to describe a policy or rule, but in a slightly indirect and oblique way -- thereby making it a bit more polite than simply saying "you must do XXX..."
Contextually, this sentence is all from the point of view of the speaker, who is your apartment agent in this case.
Putting it back together
A word-for-word annotation:
[インターネット]{internet}[の]{[possessive] }[手続き]{procedure }[は]{[topic] }[お]{[honorific] }[客]{guest }[様]{[as explicit outsider] }[の]{ [possessive] }[ほう]{side }[で]{[locative]: "on" }[取って]{take, take care of }[頂く]{humbly receive }[形]{shape, form }[に]{[directional]: "to, into" }[なります]{become}。
A literal translation:
→ Talking about the internet procedure, it is the usual way that [the speaker] humbly gets it taken care of on the side of the honored guest.
An idiomatic translation:
→ Our policy is for guests to take care of the procedure for getting an internet connection.