Use of non accredited calibration service providers

This will depend on your sector and the use of the equipment. Check first if there are regulations relevant to your clients, or perhaps requirements from your accreditation body for the type of testing and program specific accreditation.

The ISO 17025 standard does not state as a general requirement you must use an ISO 17025 accredited calibration laboratory and an in each case obtain an accredited calibration certificate. The main requirement is metrological traceability of measurement results, which includes the use of competent laboratories for your calibrations. These can be either ISO 17015 accredited or non-accredited laboratories that provide, for example, traceability to international SI units.  A reason to obtain an ISSO 17025 accredited calibration is to get the measurement uncertainty on the calibration certificate, which may be necessary for your application.  If however, the equipment has a minor impact (low risk) on the validity and uncertainty of the measurement,  then a non-accredited calibration should be suitable. For example, there are large tolerance/specifications on a pass/fail outcome. 
On the other hand, you should obtain an accredited calibration where equipment is used for critical applications, or used to calibrate or test other equipment.

For further information have a look at The article What does ISO 17025:2017 require for laboratory measurement equipment and related procedures? at https://advisera.com/17025academy/blog/2019/07/25/iso-17025-measurement-requirements-of-the-standard/ and The ISO 17025 document template: Equipment and Calibration Procedure at https://advisera.com/17025academy/documentation/equipment-and-calibration-procedure/