Yes, the shield guardian can come with
Nothing in the astral projection spell requires a soul. Instead, the spell projects an astral body, and there is nothing to suggest that constructs or shield guardians specifically lack an astral body:
You and up to eight willing creatures within range project your astral bodies into the Astral Plane
Furthermore, I see no evidence to suggest that constructs don't have souls. Souls are not a very clearly defined concept in 5th edition, so it is entirely possibly that shield guardians do not have souls, but there is no distinction one way or another. This answer about the nature of a simulacrum's soul dives into the ambiguity on the issue.
While the astral projection spell does say this:
If the cord [connecting your astral body to your material body] is cut [...] your soul and body are separated, killing you instantly.
...that soul separation could simply be a side effect of losing your astral body, and is not necessarily evidence that the astral body is itself a soul. In short, unless something were to say that constructs or shield guardian's lack an astral body, the spell works just fine on the shield guardian.
Note: The Bound trait of the shield guardian would be unusable as the amulet and the shield guardian are no longer on the same plane of existence. Otherwise, the shield guardian should work as normal.