Can I use radar service instead of DME or GPS for this ILS approach?
Tl;dr: This is really an edge case and I can’t come up with any specific wording that says “yes” or “no”. But from the following info I’d say you cannot unless ATC authorizes it. 

Since the chart specifically indicates it as required it would be illegal to use the approach. Paragraph 5-5-4 a 2 of the AIM lists under pilot responsibilities: 


  
  Follows the procedure as shown on the IAP, including all restrictive notations
  


But according to this answer ATC can approve an approach even when some equipment is inop. The answer seems to be concerning inop. NAVAID equip, rather than inop. equipment on the aircraft. Also, the section of the AIM he refers to (5-4-7d) does not include what he says. Paragraph 5-4-6d states:


  The name of an instrument approach, as published, is used to identify the approach, even though a component of the approach aid, such as the glideslope on an Instrument Landing System, is inoperative or unreliable. The controller will use the name of the approach as published, but must advise the aircraft at the time an approach clearance is issued that the inoperative or unreliable approach aid component is unusable.


This would suggest that ATC can approve an approach with inop. equipment. 

If you consider telling you when you arrive at ROLTE to be “vectoring” then ATC can give you vectors as far as the FAF. FAA order JO 7110.65W, which gives instructions for ATC indicates in paragraph 5-9-1 a 2. 


  
  If specifically requested by the pilot, aircraft may be vectored to intercept the final approach course inside the approach gate but no closer than the final approach fix.
  


Note that where you have highlighted “RNAV 1-GPS or RADAR or DME REQUIRED” is on the IAF at GQO, but not on ROLTE. This would indicate to me that you can count on any of those to get you to the IAF, but not beyond it. The word RADAR on the chart at ROLTE and EWADA indicate that those fixes are depicted on the ATC radar screen. This would certainly make it possible for them to tell you when you’ve reached the fix, but I don’t know if you can count on them to do it. And without that or DME you would have no accurate way of identifying the FAF. They may not have time to monitor you after you’ve reached the IAF, so that’s why I say it would require ATC approval.