Grammatically, the sentence is fine. My issue would be 'at last' and the point you are trying to convey. Are you trying to show it took a long time and was hard work. On its own, this phrase is more logical: They reached the North Pole because they made careful preparations ... or ... prepared carefully If you want to emphasize the length of time: They reached the North Pole, at last, because they prepared carefully ... or an alternative could be They were finally able to reach the North Pole because they prepared carefully.

Although its ok to use it this way as there is no continuation to the text but I would be careful with the use of "reached" and "have reached". Do you know the difference between the two?

It's OK. I prefer "At last they reached the North Pole due to careful preparations/planning"