Spectrum analyzer as source of electromagnetic interference
Certainly could be.  There is nothing special about spectrum analyzers.  Any piece of electronic equipment could potentially be a source of interference.  

Edit: let's work through your block diagram.


Input attenuator.  Passive component.  Shielded for dynamic range.  This will not produce any EMI.  
Input amplifier.  Active component.  Shielded for dynamic range.  This should not produce any interference unless it is perhaps badly misbehaving.  
Mixer.  Could be active or passive.  Shielded for dynamic range and LO leakage.  Could let out some of the LO frequency if not shielded correctly.  
Sweep generator.  Active component.  May or may not be shielded.  Could possibly produce low frequency EMI, especially if it is driving the coil of a YTO.  
Local oscillator.  Active component.  Shielded to prevent LO leakage.  Should not produce EMI if shielded correctly.  
Filter.  Passive component.  Shielded for dynamic range.  Should not produce EMI.  
Detector.  Passive nonlinear component.  Shielded for dynamic range.  Should not produce EMI.  
Video amp.  Active component.  Shielded for dynamic range.  Could possibly produce low frequency EMI.  


Your diagram is missing some very important components, though.  Namely, power supplies and control circuitry.  These components are less likely to be as shielded as the RF signal chain and therefore are likely to produce far more EMI.  

Here's something interesting to note, though:  user123753 mentions using a spectrum analyzer to measure it's own EMI.  If you try that on a spectrum analyzer that has a leaky LO, you will not be able to see it on the analyzer!