Note that the Yuuzahn Vong are entirely part of the non-canon Legends continuity; they do not exist in the new canon. So in the "main canon", the answer is clearly no: Palpatine did not anticipate the Yuuzahn Vong, because the Yuuzahn Vong do not exist1.
In Legends, the answer is
The novel Outbound Flight reveals that, at least as early as 27 BBY, Palpatine was aware of a threat of invasion from a group referred to as the "Far Outsiders":
"There's an invasion coming," he said. "A massive assault force of dark ships, shadowy figures, and weapons of great power, based on organic technology of a sort we've never seen before. We believe these Far Outsiders, as we call them, already have a foothold at the far edge of the galaxy, and even now have scouting parties seeking information on worlds and peoples to conquer."
"When will Darth Sidious tell [the Republic]?"
"When he's turned the Republic's chaos into order," Doriana said. "When we've built an army and a fleet capable of dealing with the threat. To announce it before then would do nothing but create panic and leave us open to disaster."
Outbound Flight Chapter 18
The Far Outsiders sound very much like the Vong we meet later, but they're confirmed as one and the same through Vergere, who's said in Rogue Planet to have left Zonama Sekot with the Outsiders:
"We came here to buy a ship from you, and to find out what happened to Vergere."
Shappa chuckled grimly. "It's all tangled, isn't it? She's gone."
"Where did she go?"
"She left with the Far Outsiders."
Rogue Planet Chapter 45
She turns up again in the New Jedi Order series as a servant of the Yuuzhan Vong.
Although we don't have confirmation that the Death Star was itself intended to be used against the Vong, we know that Palpatine was:
- Aware of the imminent threat of invasion
- Building a unified Empire for the purpose (at least in part) of repelling that invasion
Which seems to pretty strongly suggest that the Death Star was a tool for use against the Vong, even if only indirectly.
1 At time of writing, anyway; the Lucasfilm Story Group hasn't shied away from borrowing popular EU elements for new canon works, so you never know how things might change in the future. And note that some newer canon novels (Empire's End, Thrawn) indicate that Palpatine was concerned about threats in Dark Space, and that the Chiss Ascendancy was monitoring hostile somethings out there, so it's not impossible that we could see the Vong return to canon