Yes, they can if you allow them to.
Besides the PayPal payment option, which you can read about in How can I Request Payment by PayPal?, your client can pay using a credit card they have on file or with a credit card that they will provide for payment just this one time. When this option is enabled, you have the opportunity to include a "Pay Online" link in the invoice you send to a client.
Setting Up the Pay Online Option
Before your clients can pay online, you must establish a relationship with a Merchant Bank to accepts credit card payments, and that bank must permit access through one of the gateways that LeashTime supports (at this time: Authorize.net and TransFirst).
Next, LeashTime must be configured with your Merchant Account information (please read How do I set up my Merchant Account (credit card processing) in LeashTime?).
You must also select at least one credit card company from which you will accept payments. To do so, go to ADMIN > Preferences > Billing Preferences and click on "Credit Cards Accepted".
Finally, you must set "Include a "Pay Now" link in invoices" (also in ADMIN > Preferences > Billing Preferences) to "yes".
Using the Pay Online Option
When you view a new client invoice statement (in BETA Billing) and you have previously set up the Pay Online option as described above, you will notice a check box labeled "... with a Pay Now link" underneath the "Email This Invoice" button. The box is checked by default. You will also notice a "Pay Online" button under the "Amount Due" box.
If you un-check the box, the "Pay Online" button disappears. You can do this before emailing the invoice if you do NOT want the client to pay online.
If you leave the box checked, the client receiving the invoice will see the same button (or a link that says "Pay Online", if the client's email suppresses images). The client can click the link to pay the invoice and add gratuities using the card on file, or the client can supply another credit card for use just this one time.