The simplest solution is to use an existing open source GPS tracking system, and configure it for dog walking.
There is an open source package called "Trackcar" which can be downloaded, and used free of charge.
Several open source iOS, and Android apps for tracking are available for Traccar. The apps are free of charge too, and written by the same developers that developed Traccar.
Traccar can be configured for dog walking by assigning users that are customers with dogs to be walked to a group specific to them, and assigning dog walkers to the groups of the customers that they walk dog for. This will keep each customer's data, and maps separate, and allow the dog walkers access to only the customers they service. Although, if more than one dog walker share a customer, both will have access to the data.
The time, distance, and route of each walk can be tracked on the maps & trip reports.
Traccar also supports Geo-Fencing, and email / SMS alerts. So a dog owner can be notified if the dog strays from a predefined area, or if there is an emergency. The iOS, and Android apps have an SOS button in case of an emergency.
Traccar requires a Virtual Private Server (VPS) preferably CentOS 7 which is Linux.
Also note: I've been working with GPS systems almost exclusively since 2012, and have been developing software in one form or another since 1979.
Please review my profile, and contact me on the Freelancer message board so we can discuss your requirements further.