Is it possible to purify antigens from a vaccine and to separate them from the adjuvant?
The major component of a vaccine is the adjuvant, and according to Wikipedia the most commonly used adjuvants in human vaccines are aluminium salts.

Here for example is the composition of Recombivax HB, a hepatitis B vaccine produced by Merck Frosst and used in Canada (source of information is a Canadian website)


  RECOMBIVAX HB ® [hepatitis B vaccine (recombinant)] is a non-infectious subunit viral vaccine consisting of surface antigen (HBsAg or Australia antigen) of hepatitis B virus produced in yeast cells. A portion of the hepatitis B virus gene, coding for HBsAg, is cloned into yeast and the vaccine for hepatitis B is produced from cultures of this recombinant yeast strain according to methods developed in the Merck Research Laboratories.
  Two formulations are available:
   •  10 µg/1.0 mL formulation: each 1.0 mL dose contains 10 µg of hepatitis B surface antigen adsorbed onto approximately 0.5 mg of amorphous aluminum hydroxyphosphate;
   •  40 µg/1.0 mL formulation: each 1.0 mL dose contains 40 µg of hepatitis B surface antigen adsorbed onto approximately 0.5 mg of amorphous aluminum hydroxyphosphate;
  Thimerosal (mercury derivative) 1:20,000 (50 µg/mL) has been added only to the preservative-containing formulations. All preparations have been treated with formaldehyde prior to adsorption onto amorphous aluminum hydroxyphosphate. The vaccine is of the adw subtype.


As explained in   Exley, C. et al. (2010) The immunobiology of aluminium adjuvants: how do they really work? Trends in Immunology, 31:103-109:


  The vaccine preparation is primarily micrometer-sized clusters of nano-sized primary particles of the aluminium salt with which the antigen is associated by adsorption and entrapment. The avidity with which the adjuvant associates with the antigen will depend upon multiple factors, including the form of aluminium salt (usually oxyhydroxide or hydroxyphosphate), the physico-chemical properties of the antigen (including its overall charge and molecular weight), the mode of preparation of the antigen-adjuvant complex (for example, ratio of adjuvant to antigen), and the final solution pH. The latter will usually be around neutral (pH 7.0 +/-0.5), and, along with the highly super-saturated state of the aluminium salt, this will ensure that the concentration of soluble aluminium in the vaccine preparation remains below ca 2 mM. Similarly, there will be a variable proportion of antigen, often &lt;1% of the total antigen load, that is not associated directly with the adjuvant, and some of this ‘free’ antigen may also be in a complex with aluminium., what this means is that the protein antigen is not soluble in the vaccine, but rather is adsorbed and/or entrapped in the inorganic particulate material. 


What this means is that the protein antigen is not soluble in the vaccine, but rather is adsorbed and/or entrapped in the inorganic particulate material. The antigen itself has of course already been extensively purified before being incorporated into the vaccine, so the task of purifying the antigen from the Recombivax vaccine is one of releasing it from the inorganic adjuvant. The obvious thing to try first would be to isolate the particulate material by centrifugation, and then attempt to release the antigen from this in a soluble form: things to try would be extremes of pH, high salt concentrations, and detergents. SDS-PAGE would be used to monitor how much was released by these treatments.

Now the example that I gave above (Recombivax HB) is fairly straightforward because it contains a single protein antigen. Compare this to Pentacel, given as first vaccination to babies in Canada (same source as before). 


  Act-HIB ® Reconstituted with QUADRACEL ®Haemophilus b Conjugate Vaccine (Tetanus Protein – Conjugate) Reconstituted with Component Pertussis Vaccine and Diphtheria and Tetanus Toxoids Adsorbed Combined with Inactivated Poliomyelitis Vaccine.
  Each single dose (approximately 0.5 mL) after reconstitution contains:
   •  purified polyribose ribitol phosphate capsular polysaccharide (PRP) of Haemophilus influenzae type b covalently bound to 20 µg of tetanus protein 10 µg
   •  pertussis toxoid (PT) 20 µg
   •  filamentous haemaglutinin (FHA) 20 µg
   •  fimbrial agglutinogens 2 + 3 (FIM) 5 µg
   •  pertactin (PRN) 3 µg
   •  diphtheria toxoid 15 Lf
   •  tetanus toxoid 5 Lf
   •  poliovirus type 1 (Mahoney) 40 D-antigen units
   •  poliovirus type 2 (MEF1) 8 D-antigen units
   •  poliovirus type 3 (Saukett) 32 D-antigen units
   •  aluminum phosphate 1.5 mg
   •  2-phenoxyethanol (not as a preservative) 0.6% v/v
   •  polysorbate 80 10 ppm (by calculation)
   •  bovine serum ≤50 ng
   •  trace amounts of formaldehyde
   •  trace amounts of polymyxin B and neomycin may be present from the cell growth medium


This vaccine is clearly much more complex, so in this case, once the desired antigen(s) were freed from the adjuvant there would be a need to use standard protein purification techniques to separate the various contaminants.

Finally it's worth pointing out that unless you have access to large amounts of vaccine, you are going to be starting with quite small amounts of antigen (see above) so it will probably be quite a challenging purification.