Brandon Sanderson has recently confirmed that all Shards can manifest physically.
Argent: Can Shards manifest a physical body that can actually interact with the Physical Realm?
Brandon Sanderson: If they wanted to, yes.
Argent: Was that the thing that Odium did at the end of Oathbringer or was that just a projection?
Brandon Sanderson: Uhm, it starts to be really difficult to define when you're getting to these points because they generally are such massive wells of Investiture themselves that it's like, is this thing they're creating, like, they are kinda, y'know, then bending the three Realms around the like spacetime with lots of gravity so is that a projection? Is that a real thing? Does it matter? Does that definition...
Argent: Oh! At that point they are almost the same thing, right?
Brandon Sanderson: Yes, exactly.
JordanCon (April 2018)
Questioner: Other Shards, other than Odium and Cultivation, can take physical form, right?
Brandon Sanderson: Yes.
Questioner: Are there any Shards that can't?
Brandon Sanderson: No.
Questioner: Could Ruin have taken physical form in The Well of Ascension?
Brandon Sanderson: So, yes. What it really counts to be physical, for someone like a Shard, is subject to debate, but he could have done some of the things that others have done. There were certain restrictions on both Ruin and Preservation, because of the deal that they had set up, that would not have made doing that very useful. But there is possibility he could have.
The Great American Read: Other Worlds with Brandon Sanderson (Oct 2018)
Now, to be clear, while a Shard, or more specifically, a Vessel can manifest a physical presence, they can not become a physical form as that would be far too limiting.
The Physical form of a Shard (independent of the Vessel) is a solid that only does one thing or otherwise limits the power to a single type. Think Atium or Honorblades.
In physical form [the power of a Shard] is rigid and does one specific thing.
Brandon Sanderson (Oct 17, 2008)
The Coppermind article on Shards provides a good summary on the metaphysical reasoning for this and is very well sourced, so I encourage you to go look at that for the specifics that are only tangentially related to this answer.
Regarding the Vessel/Mind of a Shard (Leras, Ati, Vin, Sazed, Rayse etc), there has never been any indication that they want to manifest themselves physically. Preservation appeared as an incorporeal form fairly often in the physical world whenever there was mist about. Ruin, Harmony, and Odium (at minimum) can all project themselves into someone's mind, and do this on a regular basis. Though, this seems to take a lot of focus at the expense of their more general omnipresence.
For instance, she had learned that Ruin could either manifest in person or affect her from a distance. When its actual presence was not with her in the cell, Ruin's words were far more simple and vague.
The Hero of Ages - Chapter 57
Sanderson is fairly clear about the fact that a Shard's Power is basically incompatible with a living Physical form.
So when a person is burning metals, they aren't using Preservation's body as a fuel so to speak—though they are tapping into the powers of creation just slightly. When Vin burns the mists, however, she'd doing just that—using the essence of Preservation, the Shard of Adonalsium itself—to fuel Allomancy. Doing this, however, rips 'troughs' through her body. It's like forcing far too much pressure through a very small, fragile hose. That much power eventually vaporizes the corporeal host, which is acting as the block and forcing the power into a single type of conduit (Allomancy) and frees it to be more expansive.
Brandon Sanderson (Oct 17, 2008)
You can see when both Vin and Sazed Accended, they both lost their physical forms.
The mist swirled in a final tempest, the circular motion growing faster—yet tighter—as the final wisps of mist spun down and were pulled into Vin's body.
Vin began to tremble. She gasped, feeling the fire within her blaze hotter and hotter. It was Allomancy as she'd never known it. It felt as if she had never understood it. The power was far greater than metals, mere Pushes and Pulls. It was something awesomely more vast. A power that men had used, yet never comprehended.
And then, Vin vanished.
The Hero of Ages - Chapter 73
[Sazed] slammed his arms into the twin mists and seized the powers offered to him. He drew them in, feeling them infuse his body, making him burn. His flesh and bones evaporated, but as they did, he tapped his copperminds, dumping their entire contents into his expanding consciousness.
The Hero of Ages - Chapter 82
That said, a Vessel can give up a Shard, in which case, I assume they regain their physical form. Though whether alive or dead (like at the end of The Hero of Ages) is unclear and has not yet happened as far as we know.
Questioner: Can holders of Shards give them up voluntarily?
Brandon Sanderson: Yes, a Vessel for a Shard of Adonalsium can give up their power if they wish.
FAQFriday 2017 (Jan 2017)
(all emphasis mine)