As a store administrator, you may enable membership signup for customers in your online store (See the "Allow customers to sign up for membership" setting described here):
X-Cart 5.4.x - Store Setup > Cart & Checkout
X-Cart 5.5.x - Store Management > Cart & Checkout
If membership signup is enabled, new and existing customers can sign up for membership levels they want, provided your store has at least one active membership level.
The account signup page and the customer account details page will provide the "Pending membership" field where it is possible to specify the membership level they require before submitting the form.
Once the form is submitted, a store admin can approve or decline the user's request for membership (See further below for how to do it).
In the list of users, customers with a pending membership are marked as "requested for <membership level name>" (where instead of <membership level name>, you see the actual name of the membership level).
X-Cart 5.4.x - Users > Users List
X-Cart 5.5.x - Communication > Customers > All Customers
You can also check that information on the Store Management > Users > All Users page.
However, since a store tends to have lots of users, it's easier to find out whether any users in the store have signed up for memberships using the search toolbar on the same page (be sure to select the checkbox with the name of one or more pending memberships in the filter):
If you see a user with a pending membership request, you may want to check the user's profile and approve the membership request:
In the users' table, click on the user's email to access their profile information.
Scroll down the page to the Access information section of the user profile and locate the Membership field
Select the membership level requested by the user from the Membership field drop-down.
The membership level requested by the user displays as "Pending membership." The user's current membership level is shown in the Membership field. The "Ignore membership" option in the Membership field means the user does not have a membership level assigned.
Click Update to save the changes.