One of the problems with many of the tribes of people, or even individuals, who are mentioned in Biblical texts is that there are no extra-biblical sources. The reason is often that they were smaller players on the world stage. They existed in a really insignificant area, as we can tell by the lack of records we have in general. So this means that when dealing with people such as the Amalekites, we have to be rather careful.
Looking first at possible archeological remains, we have limited data. Beno Rothenberg, in Negeb, Archeology in the Negev and the Aravah, suggested that some of the smaller settlements in the Negev highlands may be attached to the Amalekites. He made this suggestion first in 1967, after an expedition to the area in 1966. Rothenberg would revisit the idea a handful of times, such as in 1972, and 1999, as they learned more about the site. But in 2003, he wrote an article titled Egyptian chariots, Midianites from Hijaz/Midian (Northwest Arabia) and Amalekites from the Negev in the Timna Mines. In particular interest here is the treatment of rock art that Rothenberg attributes to the Amalekites.
Ze’ev Herzog, in Enclosed Settlements in the Negev and the Wilderness of Beer-sheba, also argues that the site named Tel Masos is the same as the place that 1 Sam. 15:5 calls “the city of Amalek.” His argument is partially based on archeological remains at Tel Masos that are of the right time for Amalekite settlement, and matches up with what we know from the Biblical tradition. So what individuals like Herzog and Rothenberg are arguing is that we have a distinct group of people, settling in the areas that the Biblical tradition states, at the time they say it was happening, and generally doing what the Biblical record is saying, according to the archeological record. While the archeological record does not flesh out the name Amalekite, it isn’t a necessity, as the evidence we have suggests that is who they are.
The archeological evidence, combined with critical scholarship on Biblical texts, has led to the common view that the Amalekites were one of the groups of individuals who had settled in the Canaan area, which were later assimilated into what became the Nation of Israel. A.E. Killebrew, in her book Biblical People and Ethnicity, makes the argument (and she admits she didn’t originate this view) that the origins of Israel are a “mixed multitude.” Part of this multitude, she argues, was also the Amalekites.
So we do have tentative archeological evidence for the Amalekites, and the group of people makes sense in the larger archeological evidence that leads scholars to believe that Israel was formed by the mixture of other peoples.
That’s not to say that the Biblical record is fully accurate when it comes to the Amalekites though. Most Old Testament scholars readily admit that those works are not history as we see it today. Robert Alter, in his now classic book, The Art of Biblical Narrative, really lays down an argument of how to read the Bible (or Hebrew scriptures specifically). He argues that while the Bible contains history, it can’t be separated from theology, poetry, fiction, etc. It’s laying out history, but it is also intimately connecting it to the Jewish religion. And in part, what we are seeing is what could be called a national epic. The text is definitely biased to one side.
So most likely, they existed. Archeological evidence helps support that, especially when combined with the Biblical tradition. However, we have to be careful in what we say about the Amalekites, as the Biblical tradition wasn’t recording history as we know it today, and is definitely slanted.