We use “arrange for” when we talk about making arrangements, often when we ask someone else to deal with something, e.g.:

Please arrange for the issuance of a new invoice. (I don’t expect you to do it yourself – ask an accountant to do it.)

I will arrange for the documents to be sent to you in the morning. (I won’t send them myself – I will ask someone else to arrange a courier.)

“Arrange for” is a fairly formal phrase usually used in business or legal contexts.

It is also polite. When you ask someone to “arrange for” something, you are not directly asking the person to do it themselves. “Arrange for” implies that you only want the person you’re addressing to see that something is done, most probably by someone more junior. Therefore, when you use the phrase, you are also implying that the person you’re addressing is senior enough to have subordinates.

In contrast, “arrange” means plan, prepare, organise. E.g.:

We will arrange a meeting as soon as possible. (We will do it ourselves.)

Jane asked me on Friday to arrange a visit to the factory, but I still haven’t had time yet. (Jane asked me to do it; someone else may have asked Jane to “arrange for” it.)

“Arrange” also means to put in some form of order, e.g.:

Please arrange the furniture so the sun does not shine on my computer screen.

In this context “arrange for” is incorrect.

The difference between “arrange” and “arrange for” is similar to the difference between “do something” and “have something done”:

“arrange” = “do something”
“arrange for” = “have something done”

“Arrange for” is more formal and polite than “have something done”. E.g.:

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