If you have an ability, such as Artificers’ Spell-Storing Item, that you use when you finish a long rest, can you use that ability any time after you finish the rest or does it have to be immediately afterwards? 

"Whenever you finish a long rest, you can touch one simple or martial weapon or one item that you can use as a spellcasting focus, and you store a spell in it,"

The wording is like preparing spells you can only do it at the moment you finish a long rest

If you could do it at any time it would be worded like the spell re-fueling ring

"While wearing this ring, the creature can recover one expended spell slot as an action. The recovered slot can be of 3rd level or lower. Once used, the ring can’t be used again until the next dawn."

There isn't much argument for the timing of "whenever you finish a [whatever] rest."

Is 4 hours later when you finished the rest? No.

Simple as that.

Quote from Jegpeg >>

"Whenever you finish a long rest, you can touch one simple or martial weapon or one item that you can use as a spellcasting focus, and you store a spell in it,"

The wording is like preparing spells you can only do it at the moment you finish a long rest

If you could do it at any time it would be worded like the spell re-fueling ring

"While wearing this ring, the creature can recover one expended spell slot as an action. The recovered slot can be of 3rd level or lower. Once used, the ring can’t be used again until the next dawn."

 Good point I hadn’t considered the similarity with preparing spells.