Learn Russian: Обычно я получаю посылки утром. - I usually receive parcels in the morning.

To say who sent the parcel (a person), use от + Genitive case: Обычно я получаю посылки от друга (from a friend). If you want to say what place it came from, use из + Genitive: из Москвы (from Moscow).

Is посылки in the accusative case here, since it's the thing being received?

Yes, it is the direct object. Because a parcel is inanimate (not a living thing), its accusative plural form is exactly the same as its nominative plural form. The singular is посылка.

Why doesn't 'in the morning' use a preposition like в?

Russian uses the instrumental case on its own to express times of day. Утро (morning) becomes утром to mean 'in the morning'. You do the same for other times of day, like вечером (in the evening) and днём (in the afternoon).

Could I use the perfective verb получу here instead of получаю?

No, because the word обычно (usually) shows that this is a repeated, routine action. For habits, you must use the imperfective aspect (получать). The perfective verb получу means 'I will receive' and is only used for a one-time completed action.