Learn Swedish: Jag kan spåra försändelsen åt dig. - I can track the shipment for you.

We use åt because the speaker is doing a favor or performing a service on someone's behalf (tracking the package for them). You generally use till when physically giving, sending, or transferring an object directly to someone.

Can I drop the preposition and use the double-object word order here, saying Jag kan spåra dig försändelsen?

No, that sounds unnatural in Swedish. The double-object order (putting the indirect person object right before the direct object) is typically reserved for verbs of giving, showing, or telling, such as ge, visa, or berätta. When doing a favor for someone, you keep the preposition åt.

What is the base form of försändelsen?

The base word is en försändelse (a shipment or delivery). Since it is an en-word that already ends in a vowel, you just add -n to make it definite, resulting in försändelsen (the shipment).

Why isn't there an att before the verb spåra?

Because kan is a modal verb. Just like in English, Swedish modal verbs are followed directly by the infinitive form of the next verb without an att (to). You say kan spåra just like you say 'can track' rather than 'can to track'.