You can copy the relevant rows from 'ClientList' to 'WebDev' and other such daughter sheets using
filter() formulas like this:
=filter(ClientList!A2:B, ClientList!C2:C = A1)
The problem is that it appears that your intention is then to manually add data side-by-side with the formula results, and that exposes you to the risk of rows getting misaligned.
Lance has given a thorough treatment of the row misalignment issue and how it can be dealt with in some cases.
You could also consider an alternative way of viewing data by different types: instead of using formulas in several daughter sheets, like you are planning to do now, use filter views directly on the 'ClientList' sheet to choose which rows to show.
This approach would let you, and other users, view and edit one particular kind of data, without disturbing others, and without leaving the 'ClientList' sheet. Importantly, filter views let you edit data right there while you are viewing the filtered results.
For additional ease of use, you can insert links in the frozen section of the sheet to easily switch between filter views, instead of having to go to Data > Filter views to switch. See the Filter views example spreadsheet for an illustration.