1812, "of the nature of rivalry;" see rivalry + -ous. It seems to have been rare (not in Century Dictionary, 1891) before the later sense "given to rivalry" emerged c. 1920.
Entries linking to rivalrous
rivalry n.
"act of rivaling, competition, strife or effort to attain an object another is pursuing," 1590s; from rival (n.) + -ry. Shakespeare has rivality ("Antony and Cleopatra"), but meaning "association, partnership, equality in rank," from the secondary sense of the Latin adjective. Jonson has rivalship (1630s); rivaltry (1640s) also was used.
-ous
word-forming element making adjectives from nouns, meaning "having, full of, having to do with, doing, inclined to," from Old French -ous, -eux, from Latin -osus (compare -ose (1)). In chemistry, "having a lower valence than forms expressed in -ic."