one dangerous thing is that most conventional cancer drugs cause immunosuppression (usually bone marrow suppression) so if you're taking these cancer vaccines you're going to be forgoing conventional treatments for the duration that you're on the vaccine. But if they have a higher success rate it would obviously be worth it; even if it has the same success rate since the vaccines would likely have less side effects.
This does not appear to be correct, while one might logically conclude that a drug causing suppression would negate the effects of adoptive immunotherapy this doesn't appear to be the case.
Recent research (see below) suggests that apoptosis from chemotherapy may actually strengthen the immune response. The same group also conducted pre-clinical trials and observed that immunotherapy directly after chemotherapy was significantly more effective than either group alone or immunotherapy before chemotherapy.
If you do a literature search for adoptive immunotherapy in immunosuppressed patients you will find more studies. The only contraindication appears to be in patients with completely eliminated immune systems, which rarely happens because of chemo.
cites two most relevant papers, one of which is specifically about the strategy to replace the chemotherapeutically depleted T cells with in vitro expanded ones, post chemotherapy. It's certainly possible, though, not enough studies have been conducted, and will probably depend on which chemotherapeutic compound is administering (small molecules tend to have differential spectra of activities). Doxycycline, in mice in at least one study was shown to have no particular effect on immunotherapeutic effect. Gemcitabine, which does do some immunosuppression but doesn't activate BER - so it only affects a subset of immune cells, does potentiate immunotherapies.
