Yes there are things that are missable, but nothing unique. There are no unique inventory items that are truly missable. While there was only one missable red trinity mark in Oogie's Mansion in the original KH, in Final Mix and consequently 1.5 HD Remix, that mark was moved to not be missable. As such, there are technically no missable items or trinity marks in Kingdom Hearts 1.5 HD Remix.
There are certainly loads of other kinds of missables in Kingdom Hearts. From various chests to whole worlds(100 acre woods), the game can easily be completed without doing or seeing many things and, after certain points, some of those things become unavailable. Some were modified for Final Mix. Here's the most easily found missable guide for KH, but it does not cover these changes for Final Mix. Note:
NO UNIQUE INVENTORY ITEM in the game is missable, not if you're thorough
and complete each world.
That said, there are plenty of things in KINGDOM HEARTS (KH) with a single
or a very limited chance of obtaining as the game progresses.
This guide also lists many non-missables, but easily overlooked scenes, chests, items, trinity marks, and abilities, as well as numerous gameplay tips.