If you duplicate your mesh, split your faces by edges, unwrap and scale it all to 0 with individual origins option, you gonna have a uvmap that you can use on your original mesh.
But since the geometry nodes exist, we can transfer the uv to vertex color and use it for the mapping.
First you need to go 'Object data' >> 'Vertex colors' and create a vertex color layer.
In older versions from 3.0 you need to have two 'Attribute convert' nodes. The first is to get the uvmap and write it to a temporary variable. It's gotta be in face corner domain and the type is color. Type 'UVMap' to the first attribute and any string to the next. Second node is to get the temporary variable and write it to vertex color data. So type your attribute name to the first and select the vertex color layer below on the 'Result'. Now you can simply create a vertex color input and tie it to vector input of your texture node in the 'Shader editor'.
In 3.0, again first go and have a vertex color layer created. Add a 'Transfer attribute' node, give it the target and connect the attribute socket to 'Group input'. Now you created an attribute so press N, go to 'Group' tab, find your attribute and change the type to 'Color'. Also find the attribute on the geometry node modifier from the modifier stack and pick the uvmap as the value. Duplicate the 'Transfer attribute' node, change the type to 'Color', connect the same target and the attribute output you just did. Now link the new attribute output to the 'Group output' so you have this new attribute that you need to change its type to 'Color' and domain type to 'Face' from N >> 'Group' >> 'Outputs'. After you do all this right and pick the vertex color name as the output attribute from the modifier stack you should be able to use the vertex color layer as vector input for the mapping on your shader setup. Using the vertex color node as a vector input is not working outside cycles anymore so go to add node >> Input >> Attribute >> type the vertex color layer name and connect the vector socket to your texture.
All in the screenshot: