Not always.
While Imperial strongly encourages supervisors to make available a bursary from their research group's (departmental) resources whenever possible, or at the very least provide support to the student in applying to one or more third-party sources, there is no explicit rule which requires a student to receive a bursary from the supervisor in order to undertake a UROP research experience unless the (host) dept in quesiton has a local rule which means that a UROP cannot proceed without financial support being in place for the student, for which see news at the end of this answer).
Naturally, most students are keen to receive remuneration for their efforts on behalf of the host supervisor/research group. However, for most students it is an absolute requirement in order to cover some of the costs of remaining in or re-locating to London during Imperial's summer vacation (which is when the vast majority of UROP research experiences take place), and/or to make-up some of the funds that a student may have obtained through more regular vacation work.
Therefore, supervisors who are unable to access funds within their research group are strongly encouraged to discuss the matter with their department in the first instance, before supporting the student with any application to a relevant third-party source of funding. Imperial encourages supervisors to assist students but students must not leave it to the last moment.
For some supervisors there may be industrial contacts who may be interested in supporting the planned UROP.
All students have to take into account that competition for sources of third-party research internship funding is by nature competitive and that deadlines will apply. You need to plan ahead since third-party sources of funding may be inaccessible by the time you think of applying! Remember also that many application procedures for third party sources of funding require your supervisor to contribute in some way (e.g. a project description).
If, after applying to one or more third-party, a student still has no access to a bursary then the decision as to whether the UROP research experience takes place becomes a matter for the student (in consultation with their proposed supervisor).
If you wish to discuss these matters with the UROP Manager please feel free to email urop@imperial.ac.uk
February 2025 (amendments to previous statement in italics): To confirm that the Business School AND the following depts within FoNS, FoM and FoE have made it known to the UROP Manager that they will no longer allow UROPs to take place (to be registered) without financial support for the student having been agreed: Chemistry, Life Sciences, Mathematics, Physics, Brain Sciences, Immunology and Inflammation, Aeronautics, Bioengineering, Chemical Eng, Civil and Environmental Engineering, Computing, Design Engineering, Earth Science and Eng, Electrical and Electronic Engineering, Materials and Mechanical Engineering. These departmental rules apply to all UROPs whether the student participating is an Imperial undergraduate or not. In all instances queries should be addressed to the relevant Dept's Student Office.