First, you perhaps need to understand that Webmail is just one method to access your email account - in this case, using a web browser. Your iPad (and other devices) can also access your email account using a Mail Client App - this using one of a variety of mail protocols, of which two are IMAP and POP3.
Many Mail Providers limit the available storage (i.e., your storage quota) within your Mail Account. You also need to understand that your storage quota will be for all files and messages that are saved in your account. For example, Google offer free accounts that have a total combined storage quota of 15GB - within which all of your stored files and email messages are stored. When you exceed your quota, you may be unable to send or receive email until you have reduced the amount of combined storage that you are using.
The vast majority of modern email systems keep all your email on the email server; email client Apps download a local copy of the email stored within the server mailboxes. By contrast, webmail is merely a method to directly access your email from a web browser on any computer.
Deleting an email using webmail will delete the message from the server mailbox - and this deletion will, by design, be synchronised with all of your Mail client Apps that access the mail account. Similarly, when you delete messages using a Mail Client App, the deletion is reported to the Mail Server where the original copy is also deleted - and this deletion is synchronised with other Mail Client Apps. This mechanism greatly simplifies management of you mail account.
If you move email messages from your inbox to a separate folder within your mail account, while reducing the number of messages within your inbox, you are not reducing the storage required to store the your email. If you intend to archive messages, they must be moved/exported/saved elsewhere.
Should you choose to configure one of your Mail Clients to use POP3, you may be able to archive messages to local storage on your iPad - but you must be mindful of how you configure POP3...
The configuration of each device - in either using IMAP or POP3 mail settings - will significantly impact how your devices will interact.
If all devices are using IMAP to access your email account(s), all devices that access your mail account(s) will remain faithfully synchronised via your Mail Providers mail servers; each device will have a copy of the mail messages that remain stored on the mail server until deleted.
If instead devices are using POP3, mail messages may be downloaded (and deleted) from the mail server by the first device that accesses the mailbox. If some devices are using POP3 and others using IMAP, you see potentially unexpected results. While it is possible to use a mix or IMAP and POP3 clients, this requires considerable care with configuration if issues are to be avoided.
Consistent and reliable management of Mail can be achieved across multiple devices by ensuring that all devices are using IMAP. If you must use POP3, you must ensure that POP3 is configured on each device to "download and leave a copy on the Server" - and not "download and delete".
