mousikomisyo ni yuubinbangou to atarasii zyuusyo wo kinyuusita tokoro desu.

Because 記入する usually marks the place/item you write into with に.

• 申込書に記入する = to fill something in on an application form
• Here, the things being filled in are 郵便番号 and 新しい住所

So 申込書に means something like on / in the application form.

Because the two nouns are being treated as one combined object list:

• 郵便番号と新しい住所 = postal code and new address
• を記入した = filled in

Japanese often puts one particle after the whole list rather than after each item.

So this structure means:

• [postal code and new address] + を + fill in

Here と means and.

So:

• 郵便番号と新しい住所 = postal code and new address

This is a straightforward noun-to-noun connector.

記入する is a more formal word meaning to fill in or to enter information.

Compare:

• 書く = to write
• 記入する = to fill in/write required information in a form, document, box, etc.

Since the sentence is about an application form (申込書), 記入する sounds much more natural than 書く.

In this sentence, ところです shows the action is at a particular stage. With the past form before it, it means have just done something.

So:

• 記入したところです = have just filled in / just finished filling in

It emphasizes that the action happened very recently.

Both can be correct, but they mean slightly different things.

• 記入しました = I filled it in / I have filled it in
• 記入したところです = I have just filled it in

The ところです version adds the nuance of just now, or very recently completed.

Because the pattern changes meaning depending on the verb form before ところです:

• 辞書形 + ところです = about to do
• ている + ところです = in the middle of doing
• た + ところです = just did

So:

• 記入したところです = just filled in

The た-form is necessary for this just did meaning.

Because ところ is actually a noun.

So structurally, 記入したところです is like:

• the point/stage of having just filled it in

Since ところ is a noun, it can take です at the end for politeness.

That is why the sentence ends politely even though there is already a verb earlier.

Japanese often leaves out subjects when they are obvious from context.

So even though English would usually say:

• I have just filled in...

Japanese often simply says:

• 申込書に郵便番号と新しい住所を記入したところです。

The listener understands who did it from the situation.

It modifies only 住所.

So the phrase means:

• 郵便番号 = postal code
• 新しい住所 = new address

It does not mean new postal code and new address.

Normally, に is the natural choice with 記入する.

• 申込書に記入する = fill in on the application form

Using で would not sound right here, because で usually marks the place where an action happens, while に here marks the target location where the information is entered.

So に is the standard particle in this pattern.

Yes, that can exist, but it changes the focus.

• 申込書を記入した = filled in the application form
• 申込書に郵便番号と新しい住所を記入した = filled in the postal code and new address on the application form

In your sentence, the direct object is the specific information that was entered, so を goes with 郵便番号と新しい住所, and the form takes に.