REORGANIZE INDEX blocks other queries
Unfortunately, your understanding is a misunderstanding, and REORGANIZE requires locks for its operation. It just doesn't have as severe locking as an offline REBUILD.

Since REORGANIZE takes X locks, it has the potential to block readers.

Here's a quick demo to prove that it takes locks.

DROP TABLE IF EXISTS dbo.whyreorg

CREATE TABLE dbo.whyreorg
(ID INT IDENTITY PRIMARY KEY, junk INT)

INSERT dbo.whyreorg
SELECT TOP 1000 FLOOR(RAND(CONVERT(VARBINARY,NEWID()))*1000)
FROM sys.all_objects a
CROSS JOIN sys.all_objects b

CREATE INDEX IX_whyreorg_this ON dbo.whyreorg (junk)

BEGIN TRAN
ALTER INDEX IX_whyreorg_this ON dbo.whyreorg REORGANIZE

SELECT *
FROM sys.dm_tran_locks
WHERE request_session_id = @@SPID

COMMIT


Sample output (trimmed for space):


REORGANIZE shouldn't take locks on other tables, so I would hypothesize something in the blocked transactions in dependent on the table undergoing reorganization. Without details, it's impossible to confirm this.

Thankfully, the solution here is simple: stop reorganizing.