Learn Russian: Мы сумели выполнить это поручение вовремя. - We managed to complete this assignment on time.

The verb суметь (to manage to / to succeed) is a verb of achievement. When you successfully manage to do something, the action is naturally viewed as a completed result. In Russian, verbs expressing success, achievement, or forgetting almost always take a perfective infinitive like выполнить.

Could we use смогли instead of сумели here?

Yes, мы смогли (we could / we managed) would also work perfectly and means almost the same thing. However, сумели emphasizes that you used your skills, intelligence, or special effort to overcome a difficulty, whereas смогли can simply mean the circumstances physically allowed you to do it.

Why doesn't the ending of это поручение change, since it's the direct object?

It actually is in the accusative case here, as it's the direct object of выполнить. But because поручение is an inanimate neuter noun, its accusative form looks exactly like its nominative (dictionary) form. The neuter pronoun это follows the same rule.

I sometimes see вовремя written as two words (во время). What is the difference?