You can use a Bufferer to create a buffer with a specific width around the line, then a SpatialFilter to find all points within that buffer.

I would first take a look at the NeighborFinder transformer. This might be exactly what you need.

Your lines are your Base features, your points are your Candidate Features. Make sure to enable 'Generate list' on the attributes you need. After the NeighborFinder you can use a variety of transformers to put the information in an attribute (e.g. ListConcatenator).

@Hans van der Maarel​'s suggestion might also work, but for that method I would use a SpatialRelator rather than a SpatialFilter. You could indeed use a Bufferer to create a polygon representing the 'surroundings' you're mentioning. Then use the SpatialRelator, with the lines as the Requestor and the points as the Supplier. Make sure to enable 'Generate list' on the attributes you need. After the SpatialRelator you can use a variety of transformers to put the information in an attribute (e.g. ListConcatenator). After that, you can use e.g. a FeatureMerger to merge the information back onto the line.

Both methods require dedicated care/verification, since the definition of 'being in its surroundings' might vary, there might be overlap between 'surroundings' of different line features etc.