Yes, Elizabethan authors most certainly were familiar with Chaucer. Direct Elizabethan criticism on Chaucer survives, and he is a direct influence on several Elizabethan authors, including Shakespeare. Moreover, he is a direct source for the content of certain of Shakespeare plays, among which Troilus and Cressida is the most conspicuous case. The tale antedates Chaucer, but Shakespeare's version of it demonstrates derivation (if not exclusive derivation) from the Chaucerian telling.

By the 1590s, Chaucer's English was a bit challenging (being only marginally more accessible to the Elizabethan reader than it is to the modern reader). But it was more accessible than anything which came even shortly before, during the Late Middle English period, or which came from beyond the environs of London from a dialectal standpoint (Sir Gawain and the Green Knight or Piers Plowman, for example). And so it survives as an influence not simply because its importance within its own era cast a long literary shadow, but because it continued to be relatively readable centuries later.

Interestingly, it was not Chaucer's Canterbury Tales, but the poem Troilus and Criseyde which I mention, which was Chaucer's most celebrated work entering the Early Modern period. The predominant interest in the Canterbury Tales only asserts itself later, in the 18th century, among the English literary class.

As for earlier works of the Middle English period, influence tends to be very much indirect, where it exists at all. The accessibility of Early Middle English to Early Modern readers was very poor. And its poetical and genre conventions are foreign to the age.

Chaucer's influence casts such a very long shadow because he foretold (and helped to create) the fashions of the coming centuries (most importantly, Rhyme Royal), as much as because his words could be read as he wrote them.

Key Sources:
Spurgeon, Caroline E. F.
Five Hundred Years of Chaucer Criticism and Allusion: 1357-1900.