Yes, it is possible!
Actually jou just have to enable Visual Studio 2005 to generate .NET 1.1 applications. There is a Microsoft-Community add-in for MSBuild that enables this: MSBuild Extras - Toolkit for .NET 1.1 "MSBee". You can download it from http://msdn.microsoft.com/vstudio/downloads/tools/msbee/default.aspx (you will get a MSI file).
Once you have installed it, this 2 steps will be enough to compile webparts for Sharepoint 2003:
 Open you project file with a text editor (notepad for example), and do the following (in the MSBee help file you will find the same but with a VB project):
To import an MSBee .targets file in a C# project file, find the import element for the MSBuild .targets file. For a C# project, the line you are looking for is shown below.
<Import Project="$(MSBuildBinPath)\Microsoft.CSharp.targets" />
Immediately under this line, insert the MSBee language-specific target file. For a C# project, you will want to insert the line below.
<Import Project="$(MSBuildExtensionsPath)\MSBee\MSBuildExtras.FX1_1.CSharp.targets" Condition=" '$(BuildingInsideVisualStudio)' == '' AND '$(TargetFX1_1)'=='true'" />
Maybe you will find usefull to build the solution until no errors are found and afterwards call a script that calls to MSBuild and copies the binaries to the IIS Virtual Directory (usually located at server\\c\inetpub\wwwroot\bin).
I haven't found the way to build the solution inside Visual Studio, yet. I guess it is not that easy (from the help file: "At this time, we are not planning to add IDE support."). Comments are welcome if you know how!