Terry,
There is no “Divisional” cap badge as the Division was composed of a mix of Teeth Arms, Support Arms and Services. Since forming at the start of WW2, in its time many Armoured and Infantry regiments and battalions served in 7th Armoured Division, later Brigade so you would need to check the Order of battle (ORBAT) over the years. However, there would always be units of RHA/RA, RE, R SIGNALS, RASC, RAOC, REME, RAMC etc.
Since 2014 it has been re-titled as 7th Infantry Brigade and HQ East!
https://en.wikipedia.org/wiki/7th_In...dquarters_East
Diagrams show skeleton ORBATS of an Armoured Division in 1940 and 1944
Tim