Although there is nothing specific mentioned in the novel about the location of the family's first campsite, there are some clues that lead one to believe they probably camped on the beach close to where they landed. The text tells us that as soon as they landed, the family
set about finding a suitable place to erect a tent in which to pass the night. This [they] speedily did; thrusting a long spar into a hole in the rock, and supporting the other end by a pole firmly planted in the ground, [they] formed a framework over which [they] stretched the sailcloth [they] had brought; besides fastening this down with pegs, [they] placed our heavy chests and boxes on the border of the canvas, and arranged hooks so as to be able to close up the entrance during the night. (Wyss, 1812, Ch. 1, para. 81)
There are a couple of clues here that indicate the family camped close to where they landed. First, we are told they found their set-up spot "speedily." This would not have been the case had they had to wander the island looking for a suitable spot. We are also told they placed one of the support poles into a hole in a rock, which tells us they were probably still close to the rocky coast that is described a bit earlier in the text. Last, we are told they secured the flaps of the tent with the heavy chests and boxes they had brought with them. We are not told that they lugged those heavy items over any distance or with much effort, so again, we can assume they camped close to where they landed.
Another clue comes shortly after when our narrator tells us that he was looking for a place to pull a couple of casks ashore but "found that [he] could not get them up the bank on which [they] had landed" (Wyss, 1812, Ch 1, para. 89). He also tells us that "Fritz, leaving a loaded gun with me, took another himself, and went along the rough coast to see what lay beyond the stream; this fatiguing sort of walk not suiting Ernest's fancy, he sauntered down to the beach, and Jack scrambled among the rocks searching for shellfish" (para. 88). This stream is mentioned again just a few paragraphs later when Fritz returns to the campsite and tells everyone that he has ventured beyond the stream and found a lovely spot for a possible more permanent campsite (para. 118). Our narrator has not gone very far, and Fritz leaves the campsite to walk the coast. Because we are also told that Ernest doesn't want to walk so far, he saunters down to the beach, which is indicative of the beach being within easy walking distance. This is reinforced when our narrator tells us that Jack, too, the youngest, is crawling around the rocks and the tidal pools. These all point to the family's campsite being very close to where they landed.
Finally, early in Ch. 2, we are told the family wakes after a comfortable sleep and one of the first orders of business is to further explore the island. To that end, it was decided " that, as soon as [they] had breakfasted, Fritz and [our narrator] should start on an expedition with these objects in view, while [his] wife remained near [their] landing- place with the three younger boys" (Wyss, 1812, para. 3). Here, we are told as clearly as it gets that the campsite is close to where they landed.
So although we are never told expressly where the family first made camp, there are ample clues that allow us to assume they stay close to where they landed, a safe distance from the water's edge, on the beach side of a stream.