I don't want to rain on your parade, but there are a couple of issues with this design:
1. Why spend $$$ and time on building a vacuum chamber for a mediocre LM399? The LM399 will typically return to a value within specs even after a longer time off. Just don't cool it down to very cold temperatures. Rather build a tiny 'oven' around it to keep it above lets say 15 °C (Kleinstein already suggested this).
2. The vaccum feedthrough will likely be kovar in a glass seal. Since the vacuum chamber obviously needs to be bigger than the diode and is likely hand-made, the temperature gradient in the feedthrough will most likely be bigger than from the diode substrate to the the PCB. So the thermal EMF will be a lot worse in this configuration.
If you want to build something around the LM399, I would recommend you buy a batch of LM399 (like 5-10) and bring about 6 months of time during which you monitor those diodes in a burn-in circuit (or the final circuit if you just want 7V out as the BOM cost ist relatively low). Burn them in for like 1000-2000 h and toss out any stinkers, that do crazy stuff like excessive popcorn noise or sudden 1-2 ppm jumps. Then subject the rest (probably half of them will get to this point), to a test, where you power cycle them to see how they behave. Again, toss those, that show unpredictable behaviour.
This is probably the best you can get with the LM399, maybe low single digit ppm uncertainty over a year. By the time you done with that, the market value of that diode would likely be around $300, due to the equipment and man-hours spent. At that point, you may consider going for the LTZ1000, ADR1000 or even wait for the ADR1001, which is about the same price range (dunno about the ADR1001, though).