Yes - as long as they meet the other requirements for doing so.
The SRD and the Magic Item Compendium (pp. 219-220) summarize the rules for using magic items, and don't include any restrictions that would prevent a character who was an intelligent item from doing so. The only restrictions listed are those based on specific capabilities - for instance, in order to use an item whose activation item is Command Word, you have to be able to speak.
Since the intelligent item class you're using is explicitly able to speak and manipulate objects, most forms of activation are going to be possible. The major exception to watch out for is missing body slots. The Magic Item Compendium (p. 219) has rules for body slots on creatures with non-standard body shapes (e.g., creatures without legs don't get a feet slot, etc.). Depending on what kind of item the character is, several of these are likely to apply.
This character would also be unable to use potions, since...
A creature must be able to swallow a potion or smear on an oil.