Google Tag Manager
MakePlans can automatically send visitor data to your Google Tag Manager account. All you need is to enter you GTM ID. To find your container ID see https://support.google.com/tagmanager/answer/6103696?hl=en. Note that you only need to enter your container ID - not the code snippet (JavaScript).
Under 'Setup -> Booking site' you will find an input for entering your GTM ID.
Track conversions with GTM
When a booking is completed MakePlans will trigger the following event: 'booking_completed'.
You can then create a new trigger in your GTM workspace. This trigger should be of type "Custom Event" and you should filter so the trigger only occurs for Event equals 'booking_completed'. You can then use this trigger for the appropriate tag for a conversion.