Learn Dutch: Kun jij me de software laten programmeren? - Can you let me program the software?

We use laten because this is about giving permission or delegating a task. The verb laten is used for conscious, voluntary actions. If we used doen, it would imply an involuntary physical or emotional reaction, which doesn't fit the conscious action of programming software.

Why are both laten and programmeren pushed to the very end of the sentence?

Because this is a question starting with the modal verb kun (can). In Dutch main clauses, the conjugated verb comes first in a yes/no question, and all other verbs are sent to the end. The causative verb laten groups tightly with the bare infinitive programmeren.

Why does me de software come before laten programmeren, unlike the English translation?

In English, the causative verb (let) is separated from the main verb (program) by the object (me). In Dutch, the verbs at the end of the sentence (laten programmeren) usually cannot be split by objects. Both objects (me and de software) must be placed in the middle of the sentence.

Could we use mij instead of me here?

Yes, you could use the stressed pronoun mij if you want to heavily emphasize who gets to do the programming. The unstressed me is used here because the focus is on the general request, rather than contrasting you with someone else.