Is vigenary a real word?

Probably not. Vigesimal is a word, but the series primary, secondary, tertiary… seems to cover only numbers from 1 to 10 and 12:

https://en.oxforddictionaries.com/explore/what-comes-after-primary-secondary-tertiary

Vigenary is marked by the OED as rare, but the alternative vicenary is not so marked.

Vicenary ... adj. ‘Belonging to twenty’ (Bailey, 1727); based on the number twenty. Cf. vigenary adj.

• a1831 G. Peacock Arithm. in Encycl. Metrop. (1845) I. 371/1 Such a practice would naturally lead to the formation of a vicenary scale of numeration.
• 1834 Penny Cycl. II. 337/2 In France the scale from 60 to 100 is strictly vicenary (by twenties).

The root of both 'vigenary' and 'vicenary' is from Latin vīcēnārius, in turn from vīcēnī, the distributive form of vīgintī ('twenty').