Is it illegal to fly a Cessna 172S under day VFR with an inoperative vacuum gauge? Let's say my vacuum gauge is not working on my C172S and I want to go day VFR flying.  There is no MEL.  14 CFR 91.213 details procedures for inoperative instruments, and I think the following paragraph applies to my operation, so let me go through these one by one:


  (d) Except for operations conducted in accordance with paragraph (a) or (c) of this section, a person may takeoff an aircraft in operations
  conducted under this part with inoperative instruments and equipment
  without an approved Minimum Equipment List provided - 
  
  ...
  
  (2) The
  inoperative instruments and equipment are not -
  
  (i) Part of the VFR-day type certification instruments and equipment
  prescribed in the applicable airworthiness regulations under which the
  aircraft was type certificated; (N/A, vacuum gauge not part of TOMATO FLAMES)
  
  (ii) Indicated as required on the aircraft's equipment list, or on the
  Kinds of Operations Equipment List for the kind of flight operation
  being conducted; (not sure here*)
  
  (iii) Required by § 91.205 or any other rule of this part for the
  specific kind of flight operation being conducted; (N/A, not conducting flight at night or IFR)
  
  (iv) Required to be operational by an airworthiness directive; (N/A, let's assume no applibale ADs)


*This is interesting language: "Indicated as required on the aircraft's equipment list".  AFAIK, to determine if the vacuum gauge fits this description, I should consult the equipment list in the POH, which reads:



Ok, so I now assume any instrument with a number ending in "-R" is "required" for the purposes of complying with the above regulation.  Locating the vacuum gauge in the equipment list, I see it indeed has a suffix "-R", indicating it is a "required item for FAA certification":



Thus I am led to conclude that the vacuum gauge is "indicated as required on the aircraft's equipment list", and hence continuing my flight would be in violation of 14 CFR 91.213.

Is my reading of the regulation and my interpretation of "required item" accurate here?
Your reading is correct: a Cessna 172 with an inoperative vacuum gauge is not airworthy.
Type Certificate Data Sheet No. 3A12, Revision 85 shows how Cessna 172 models meet FAA airworthiness requirements. Section XII, “Model 172S, Skyhawk SP, 4 PCLM (Normal Category), 2 PCLM (Utility Category), Approved May 1, 1998,” contains

Equipment
The basic required equipment as prescribed in the applicable airworthiness regulations (see Certification Basis) must be installed in the airplane for certification …
NOTE 2: Pilot’s Operating Handbook and FAA Approved Airplane Flight Manual (POH/AFM):  part number 172SPHUS-00 (or later approved revision) is applicable to the Model 172S. The airplane must be operated according to the appropriate POH/AFM …

The Comprehensive Equipment List in Section 6 of the POH/AFM gives the Combination Vacuum Ga[u]ge/Ammeter the R suffix, meaning required.
FAR 91.213(d)(2)(ii) reads

(d) Except for operations conducted in accordance with paragraph (a) or (c) of this section, a person may takeoff an aircraft in operations conducted under this part with inoperative instruments and equipment without an approved Minimum Equipment List provided -
  (2) The inoperative instruments and equipment are not -
    (ii) Indicated as required on the aircraft's equipment list, or on the Kinds of Operations Equipment List for the kind of flight operation being conducted;

But the equipment list does indicate the gauge as required, so this exception does not apply.
To fly the airplane to a repair station to fix the vacuum gauge, apply for a Special Flight Permit (or “Ferry Permit”) with your geographic region’s Flight Standards Office, as provided in FAR 21.197.