To clarify, the ASA is on the customer's site with connected interfaces in VLAN1 and VLAN100? You then have, I presume a site-to-site VPN or remote access VPN to the ASA from your end? Is this correct?
If so, for the VPN are you encrypting the interesting traffic of 192.168.100.0/24 from your blocks?
On the ASA you should have an ACL, which should be applied to a crypto map for example:
access-list client extended permit ip object-group client-networks object-group our-networks
crypto map outside-map 20 match address client.
The object-groups (client-networks) should have the objects with the necessary networks in i.e. 192.168.1.0/24 and 192.168.100.0/24. Our-networks should contain your netblocks for instance 10.10.10.0/24
If the VPN is fine and VLAN 100 is not connected to the ASA, you will need a route from the ASA to something which has a layer3 interface on both VLAN 1 and VLAN 100. If the ASA is connected to VLAN1 and a layer3 switch has an interface on VLAN1 and VLAN100, point a route from the ASA to 192.168.100.0/24 to the VLAN1 IP address on the layer3 switch.
It might be worth sharing some configuration of the ASA, if you require further help.
Joel