I note that the blind bard himself is a little chary about exactly describing Charybdis:
But the other cliff, thou wilt note, Odysseus, is lower—they are close to each other; thou couldst even shoot an arrow across—and on it is a great fig tree with rich foliage, but beneath this divine Charybdis sucks down the black water. Thrice a day she belches it forth, and thrice she sucks it down terribly. Mayest thou not be there when she sucks it down, for no one could save thee from ruin, no, not the Earth-shaker.
For on one side lay Scylla and on the other divine Charybdis terribly sucked down the salt water of the sea. Verily whenever she belched it forth, like a cauldron on a great fire she would seethe and bubble in utter turmoil, and high over head the spray would fall on the tops of both the cliffs. But as often as she sucked down the salt water of the sea, within she could all be seen in utter turmoil, and round about the rock roared terribly, while beneath the earth appeared black with sand; and pale fear seized my men. So we looked toward her and feared destruction
Note that there's a lot of talk about Charybdis' actions but she's mostly hidden by the roil of water. Why is this important enough for these huge quotes? Two reasons:
First, it is never wrong to quote from the Odyssey.
Second, there is no direct statement that she is sessile, just that she likes to hang around this narrow, shallow waterway.
Now here's the deal. I can't justify that huge of a creature anchoring itself permanently to one spot. What I can see is a creature like this:
A cross between a giant squid and a humpback whale
The gulp/chomp/spit feeding cycle reminds me of how humpback whales feed. But those creatures like deep water. To navigate shallow areas, Charybdis can crawl around on the tentacles. She does this betimes in order to find the best feeding grounds.
In ancient days, Charybdis fed on all those clouds of fish that hang around reefs and such. But once the Bronze Age began and those foolish Greeks started sailing around her haunts so ... yummily, she changed her dietary habits.
Update: To clarify, Charybdis departs from a krill-eating whale feeding pattern, preferring larger fish, which travel in schools. She maneuvers to where conditions are favorable (shallow sunlit water and subtle factors beyond our ken) and settles until a school comes within range. Then snap! More recently, of course, divine Charybdis has learned that flimsy galley ships are also full of tasty snacks, and their travel patterns are quite predictable -- narrow waterways, approaches to ports, and anywhere Scylla isn't. ;D
I hope this helps! Many Akaioi perished providing the research necessary for this answer.