"Is it necessary to be conversant in the Hebrew tongue in order to be accepted into the Jewish community via conversion? That is, can I convert without learning Hebrew?"
The short answer is, "Yes, you can convert."
Introduction to Judaism courses, offered around the country and taken by many prospective converts, usually enable students to "decode" Hebrew - that is, to read the sounds so that at least they can follow along with the liturgy. Of course, it is our hope that converts, as well as born Jews, will learn more Hebrew than that so that they can at least understand what the basic prayers mean - but, of course, that takes time.
Now, if you ask the question because you are aware of having challenges with language acquisition, I suggest speaking with the clergy person sponsoring your conversion. I am sure they would be happy to work something out with you.
Otherwise, I encourage you to jump right in to the excitement of learning this wonderful ancient/modern language. The fact is, Judaism is more exciting the more you understand the mother tongue, because much of our Torah interpretation is based on linguistics - but don't despair if you're not a scholar from the get-go!
Find an Introduction to Judaism class near you.