This could be done exactly as you describe with an external (paid) service like Zapier (and I can provide further help on that if desired). However, it cannot be done with Airtable’s native functionality, as such.
There are other ways to accomplish the same effect, though not with the actual moving of a record.
One idea is to split the process up between 3 tables:
1 - Containing manually entered “Leads” - linked to Table 3
2 - Containing “cold leads” from the web form - linked to Table 3 if you want to move it to lead
3 - Your Sales Pipeline, main working sheet
In other words, Tables 1 and 2 are just funnels for getting the leads you want into Table 3, your Sales Pipeline and main working table. You can then use “Lookup” and “Rollup” fields to pull information you need from Tables 1 & 2 into Table 3 automatically, and use “Formula” fields to combine information from disparate “Lookup” and “Rollup” fields so you can hide the ones you don’t need to see.
If you’d like further help implementing that, or another idea, ask away
**If you are coming from a “spreadsheet” environment, as your usage of the term ‘sheet’ suggests, you’ll find that you need to think differently about using Airtable. While Airtable does lack some of the functionality that is really basic and simple to use in spreadsheets, it replaces that functionality with others that are very powerful as a sortable/filterable/linkable relational database. Once the idea of linking records and using “Lookups” and “Rollups” clicks for you though, you’ll see that there are endless possibilities.