Few destroyer classes of WW2 were able to reload their torpedoes whilst at sea, below I have listed several that can.

Several destroyer classes of the IJN were able to reload their torpedo tubes, at least partially:

The Fubuki class had 9 torpedo tubes in 3 triple mounts, carrying 6 reloads, carried in containers built into the super-structure of the ship:

(The panels in the centre of the image are the doors for these containers)

Torpedoes would be winched out of these on to a crane structure, and swung around to be loaded back into the tubes from the front.

Later IJN destroyer series based off of the Fubuki class, such as the Shiratsuyu, Kagerou and Yugumo classes changed this for 8 torpedo tubes in 2 quadruple mounts, carrying a full reload for both launchers in deck mounted containers:

These were far more convenient than the previous design of containers in the superstructure of the ship, as with the deck mounted design, the torpedo tubes simply had to be traversed so that they faced the container, then the torpedo could be (relatively) simply slid out of the container directly into the front of the launcher.

The IJN Shimakaze however, (the only ship of her class) carried 15 torpedo tubes, in 3 quintuple mounts, due to the size (and weight) of these mounts, there was no space left for reloads, so she relied on simply carrying a large amount of torpedoes to start off with:

Another class of IJN destroyer, where the reload was especially helpful, was the Akizuki class destroyer:

These ships were designed as AA destroyers, mounting 8x 100mm dual purpose guns in 4 twin turrets.

Due to the amount of main guns carried, the Akizuki class only had room for a single quadruple launcher in the centre of the ship, giving a total of just 4 torpedo tubes:

A full reload for this launcher of 4 torpedoes was carried (visible to the right of the torpedo tubes, underneath the AA mounts). These were reloaded in the same way as the other IJN quad mount torpedo tubes. This reload meant the Akizuki class was capable of firing 8 torpedoes total, matching the base torpedo salvo of one of the earlier IJN destroyers.

Several Soviet destroyers also carried reloads for their torpedoes, notably the Leningrad class destroyer leaders:

These carried up to 10 reloads for their 8 torpedo tubes, but unlike the destroyers of the IJN, these had no dedicated containers, being simply carried on the sides of the vessel (just visible underneath the boats on the image, the silver gleam), so would not necessarily have been carried as standard. They would also be slower to reload, being further from the launchers and lacking dedicated reload gear.

In general for destroyers of this era, torpedo reloads were relatively uncommon, due to the size and weight of the weapons in comparison to the ship.

This was especially an issue with the IJN vessels, as their larger torpedoes (610mm as opposed to the more standard size of 533mm) meant their weapons were especially heavy, at 2.7 tonnes per torpedo. So a full 8 torpedo reload for a Kagerou class would add an additional 21.7 tonnes of weight to the vessel. Making it even more top heavy than the IJN destroyers already were.

Torpedoes were typically used sparingly and from long ranges, with the exception of the listed ships (and potentially others I am not aware of) once a destroyer had fired its torpedoes, it would have to return to a port in order for them to be reloaded.

The images used are screenshots from the game “World of Warships”. Whilst this game may not be hugely historical in terms of gameplay, it does have very detailed 3D models of the warships of this era, so the images and information above are accurate to the real ships.