In this context, 

'People you are social distancing with' 

is short for 

'People you are practicing social distancing with'. 

The noun phrase 'social distancing' is now being used as a verb, rather than as a direct object of 'practice'.

That is, 'to practice social distancing' (Verb + Direct Object Phrase) has been shortened to 'socially distancing'. 

Here, 'practice' means 'to carry out or perform (a particular activity, method, or custom) habitually or regularly' ([Lexico][1]). 

The workbook wants the kid to draw a picture of the people she/he is habitually or regularly *(practicing) social distancing **with*** as a way to recall these days, people, and activity in the future--and especially the people ***with whom*** she/he has been (practicing) social distancing.

For 'social distance' as a verb, see ([MacMillan][2]), whose definition happens to be 'to practise social distancing', using the UK spelling of the verb 'practice/practise'. 


  [1]: https://www.lexico.com/en/definition/practice
  [2]: https://www.macmillandictionary.com/dictionary/british/social-distance