No, an investor who wants to do a 1031 exchange cannot take partial ownership interests in a company that takes title to real estate (the typical syndicate structure). Such interests are considered partnership interests by the IRS and are specifically ineligible for 1031 exchange.
The 1031 investor would have to take direct title to his or her proportionate share of the real estate. Your syndicate could own the rest. Unfortunately for syndicators, profits get split amongst the TIC owners at the property level, so you can’t earn any distributions from the portion of the property owned by the other party, except perhaps an asset management fee. Additionally, you would have higher legal fees, as this structure would require all of the usual syndication documents plus a tenant-in-common agreement and an asset management agreement.
Unless the 1031 investor is bringing a substantial amount of money (e.g., 50%) that you don’t think you can raise from other passive investors, it’s not really worth it for you to coordinate the exchange, as you could be giving up a substantial portion of your earnings.