I'd have to fairly say that this is quite a trophy to miss. You end up getting Skill Posters from missions in Legend Log as well as Dream Log, so if you're progressing through either of them, chances are you will end up acquiring more than enough. If anything, Dream Log seemed to provide a lot more than Legend Log, but that's also because of the different order you'll end up meeting some named characters in DL compared to LL.
Do note that this is asking for Skill Posters and not Skills. Skills might take you a bit longer, but for me, this unlocked while I was partially through Chapter Two and had done two islands in Dream Log. If you generally unlock one to two posters per mission, you could expect to get this within 10 missions, give or take any variation.
Possible Skill Posters early on, and where to get them:
Luffy, Zoro, and Axehand Morgan: Prologue
I believe Buggy and Alvida are from Ch. 1, Ep. 1 (I think Nami is from Arlong Park?)
Usopp and Kuro from Ch. 1, Ep. 2
Sanji and Don Krieg from Ch. 1, Ep 3
?Nami?, Arlong, and Hatchan from Ch. 1, Ep. 4
Smoker and Tashigi from Ch. 1, Ep 5
Chopper, Laphan, and Wapol from Ch. 2, Ep. 1
Mr. 1, 2, and 3 are from Ch. 2, Ep 2 (maybe Robin is from this one or the one after, Chapter 2, Episode 3?)
Bellamy from Ch.2, Ep.3 for sure, as well as Burgess.
You should already have Ivankov and Law's SPs so you might easily get this from completing Ch.2, Ep.2, if not sooner due to involvement from Dream Log. If anyone could confirm Nami and Robin's SPs, I can correct this. I didn't pay attention to where I got what (though I did get Ace, Minotaurus, Monet, and Hody early by doing the first three islands in Dream Log).