I need to setup a webhook on Salesforce Professional Edition. For example, if someone creates an account, I want the request to be sent to an external server.
Is this possible?
No, the Professional Edition (e.g. Lightning Professional) does not have workflow or automation, so you can't generally set up code to call external systems. I do believe that at one point workflow and automation was available as a paid upgrade for Professional Edition, but I do not see that as an option in the current product comparison chart. You might want to contact your account executive to confirm what your options are.
The Professional Edition has Process Builder as the only automation tool .The reference for this is below
You can spin a sample professional edition to see what you can achieve by simple signup
Also on the possibility of what you need is definitely doable .You will need to create a own managed package solution and pass the security review and list on an appexchange as ISV partner .
Check the below link.Also to push free apps on appexchange it does not cost you anything .Also security review charge is free for free applications .