There doesn't seem to be any clear explanation for why modern French doesn't use avr- as the future/conditional stem of avoir. Savoir shows the same AV/AU alternation. In contrast, we do see -vr- in the future/conditional stems of a number of -evoir verbs, such as devoir and recevoir.
The forms [so] and [o] which replace the future/conditional roots of the French verbs savoir 'to know' and avoir 'to have' (savra/avra, savroit/avroit > saura/aura, saurait/aurait) are of unexplained origin (see Maiden 1992). Possibly they reflect a sporadic or dialectal sound change...
("Morphological Innovation", by Martin Maiden, in The Cambridge History of the Romance Languages, Volume 1 (2011), p. 265)
Something I found out that may be related in some way: the verb boire seems to have lost -v- not only in its future/conditional stem but also in its infinitive form. Wiktionary says that in Old French, we can see the infinitive form boivre and the future/conditional stem bevr-.