Either will be understood, and can be correct, but in my opinion the inclusion of the word "with" has a connotation of a more general problem.
There is a problem sending this form.
This probably implies that there is a specific problem with the form, that prevents it from being sent. Perhaps you forgot to include your last name, and the web site is rejecting it as invalid.
There is a problem with sending this form.
This could suggest a more general problem, which is preventing sending the form, or which may result from sending the form. It may not be a problem with the form directly, but perhaps your Internet connection is not working. Or perhaps your computer has caught fire. Or it may be that if you submit the form, a problem will arise--perhaps your employer will discover you are looking for work elsewhere.
with could also suggest a personal or moral objection, rather than a practical, problem:
I have a problem with going to strip clubs.
She had a problem with the way he sang.