All the members of Orion are within the Milky way, and some of them look pretty close to a neighbor. And could the bow* be an effect similar to the Radcliffe Wave?
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The three stars in Orion's belt, along with Sigma Orionis and the Orion Nebula cluster, along with several other young clusters and star forming regions in the central Orion region are all part of the Orion OB1 association at a distance of 350-420 pc.
A review of Orion OB1 can be found in Bally (2008). The three belt stars are probably a similar age and distance and form the brightest stars of the OB1b sub-association. Measured distances to these stars are individually rather uncertain, but better estimates come from the low-mass, co-moving stars around them. It is therefore true to say that the individual distances to each star are not known well enough to be sure that they are close together in distance.
The other bright stars in Orion - Betelgeuse, Rigel, Bellatrix, Saiph are significantly closer than this by 150-300 pc and probably not at a common distance. It has been speculated in the past though that these stars and the Orion OB1 association are part of one, large star forming complex, extending 200-300 pc along the line of sight (Bouy & Alves 2015).
The recent paper, claiming the discovery of a "Radcliffe wave" gas structure in the solar vicinity, does include the young stars in Orion as part of this structure.
So to spell it out: The belt stars are within 70pc of each other in terms of distance. If they are at a similar distance of 400pc, their angular separation on the sky of around 3 degrees, translates to a tangential distance separation of about 20 pc. Other members are more separated than this.
Based on links on this page, the closest visible star in Orion (that is, the region of the sky called Orion) is Pi3 Orionis (π3 Orionis), a.k.a Tabit, a.k.a al-thābit, a.k.a. Zhāng Qí Liù. This is the point where the arm touches the shield/bow. That page says it is 26.2 light-years away; Wikipedia says its 26.32 ± 0.04 light-years or 8.07 ± 0.01 parsecs away. Both call it a main sequence F6 star (F6V). In any case, it's a very nearby star and much closer, and smaller, than the bright stars of Orion that you are probably thinking of, which are all hundreds of light-years away.
Based on other links on that page, the (very dim) known stars in Orion closer than π3 Orionis are:
(Those 1st 2 MIGHT be 2 names for the same star, based on my attempts to look them up, even if those who made the page obviously didn't think so.)
Almost as close is the visible binary star Χ1 Orionis (Chi1 Orionis) (the tip of the club/arm), which is 28.3 ly away according to that page and 28.26 ± 0.07 ly or 8.66 ± 0.02 pc away according to Wikipedia.
If we limit ourselves to only tallking about Betelgeuse, Rigel, Bellatrix, Mintaka, Alnilam, Alnitak, Saiph and Meissa, then the answer is probably Bellatrix (the top shoulder on the viewers right, as opposed to Betelgeuse, which is on the other side). Wikipedia cites this paper* for it estimated distance of 250 ± 10 light-years or 77 ± 3 parsecs away.
That being said, distances are one of the hardest things to measure in astronomy, and, for example, I know that Betelgeuse used to have huge uncertainties in its distance before it was recently narrowed down to being about 25% closer than the older most likely distance. The current distance is 458~597 ly or 153~195 pc, the last range coming directly from the October 2020 paper.* This makes it probably the second closest of those bright stars in Orion I mentioned.
As a final note, it's important to remember that constellations are just stars that happen to be in about the same direction from Earth's perspective. They aren't actual collections of stars, even if there may be actual collections of stars in them, like how several (but definitely not all) of the stars in Orion are suspected to be related to each other. I didn't know anthing about the Radcliff wave until I looked it up just now, nor do I know if the bow/shield ASTERISM (i.e. pattern people see in the sky) might PARTIALLY correspond to anything physical, and I'm not an astronomer.
*van Leeuwen, F. (November 2007). "Validation of the new Hipparcos reduction". Astronomy and Astrophysics. 474 (2): 653–664. arXiv:0708.1752. Bibcode:2007A&A...474..653V. doi:10.1051/0004-6361:20078357. S2CID 18759600.
*Joyce, Meridith; Leung, Shing-Chi; Molnár, László; Ireland, Michael; Kobayashi, Chiaki; Nomoto, Ken'Ichi (2020). "Standing on the Shoulders of Giants: New Mass and Distance Estimates for Betelgeuse through Combined Evolutionary, Asteroseismic, and Hydrodynamic Simulations with MESA". The Astrophysical Journal. 902 (1): 63. arXiv:2006.09837. Bibcode:2020ApJ...902...63J. doi:10.3847/1538-4357/abb8db. S2CID 221507952.
In this Wikimedia animation of Orion's appearence from Earth evolving over time from the distant past to the distant future, you can see π3 Orionis moving rapidly towards Bellatrix (not actually towards Bellatrix in space, just towards being roughly between it and Earth). Bellatrix and the other bright stars barely move scross the sky because they are much further away, and so would have to move much faster through space to move the same speed across the sky. The other rapidly moving stars are probably other nearby stars.