No. There is no need to supply a mobile phone number when you make an online payment. However, if you wish to pay using a credit card or debit card, there is a need to supply a phone number which could be a mobile number, a home number or a work number.
A phone number is only required if you have chosen the option “Pay with a credit or debit card”. In this case, you have to provide your contact details including full name, billing address, a phone number and an email address. This is common practice for any credit card or debit card payment over the web. The reason given by PayPal for requiring a phone number is “We’ll use this to contact you in case there’s a problem with your account or purchase.”
On the other hand, if you choose the option “Pay with my PayPal account”, you can simply login to your PayPal account to complete the payment. In this case, you will be able to choose your funding source which could be your PayPal account, your linked bank account, your credit card or your debit card.