You most probably can if you were wildshaped
This answer differs depending on if you are polymorphed, or if you are wildshaped. If you are polymorphed, you don't get to keep your class features, you only keep your alignment and personality:
The target’s game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains its alignment and personality.
Game statistics includes your class features, so these are overwritten. When you get 0 to zero, you revert to your original form. There is no conflict. Once you are back to your original form, you get your hit points back, and you get the feature back. Only if you took enough damage to reduce also your original form to 0 from spillover you could then use the feature to take an extra turn. Otherwise, your have more than 0 hp, and at no point had both the feature and 0 hp.
If you are wildshaped, you retain the use of your class features (p. 67, PHB). Wild Shape says:
You automatically revert if you fall unconscious, drop to
0 hit points, or die. (...) You retain the benefit of any features from your class, race, or other source and can use them if the new
form is physically capable of doing so
The feature says:
If you take damage that reduces you to 0 hit points, you can use your reaction to delay falling unconscious, and you can immediately take an extra turn
Both are not exclusive. You briefly fall to zero hit points, and automatically revert to your normal form. Since you fell to 0 hit points, you also can take your reaction to take an extra turn.
Timing of simultaneous effects
Technically, as both these effects would want to happen at the same time, the DM resolves the order, or the active player, if you use the optional rule in Xanathar's Guide to Everything for simultaneous effects.
But it really does not matter -- they resolve both in an instant, and then you are in your normal shape, if you first take the extra turn, then revert, or if you first revert then take the extra turn: you end up in your normal form either way with however hit points it has after spillover, and then take the extra turn in your normal form. The features says "If damage reduces you to 0 hit points", not "If you are at 0 hit points", and damage did reduce you to 0 hit points while you had the feature, even if after reverting first, you then are not at 0 any more.
Not falling unconscious
This answer observes that the feature can be read to substitute falling unconsious, and when you revert form you would not be falling unconsious unless spillover damage took you all the way to 0 in your native form, so there would be nothing to replace. However, the delay is not in response to falling unconsious, it is in response to the trigger of "when you take damage that reduces you to 0 hit points", so you do not need to fall unconsious first. Further more, the way it is written, it is gramatically not clear if you even need your reaction to take an extra turn. The two things that happen when damage reduces you to 0 hit points are:
you can use your reaction to delay falling unconscious, and you can immediately take an extra turn
They are listed independently, separated as two subclauses by a comma. This does not say "you can use your reaction to delay falling unconsious and to immediately take an extra turn". While I personally think that is probably what is meant, that is not what is written. So this in the end must be decided by the DM. Overall, this objection carries enough weight that a DM could use it to rule this does not work. (I personally think that would be a bit cheap for a character using a level 18 class feature, but that is a matter of taste).