Obviously even in the context of Hebrews-11(6), the choice of English, here "is" or "exists", is/exists to be decided by the translators on the basis of the passage's tone, as perceived to have been spoken:
- by which was in the passage as a whole;
- as well as by whose words and phrases had required emphasis;
- and whereby to support their perception of tone or nuance. .
As you see by my own example above, the word "exists" weighs, in connotation, much heavier in emphasis and tone, conveying deliberate purpose and function: wherein something exists to peform a purpose and fulfil a function. If it is to do so, it requires a trigger. Otherwise it sits ready, existent with latent or dormant function. .
In this verse, the speaker seems to be defining faith in terms of our need for its source, but where "seeks" (ἐκζητοῦσιν) can equally be translated as "needs". If the latter, then the emphasis could equally be intended for the conjunction (ὅτι) immediately preceding "is", thus apparently justifying the use of "exists". .
However, if the conjunction is intended to mean "Who" then it resonates with Exodus and YHWH's "I am Who I am". Nonetheless, there was no conjunction originally in the Exodus hebrew. I would love to check the manuscripts for Hebrews, but we apparently do not have the hebrew original of the Letter to the Hebrews. If by St Paul, who according to N T Wright (Q/A on Teams, Oxford 2020) was equally fluent in Hebrew and Greek, I would defer my conclusion by this.
He alas would have most likely given this letter in hebrew, originally, and thus we await the archaeologists' retrieval.