Short answer: No, Plugins can not alter the in-game map or its icons (they are not exposed to Lua).
Long answer: No, but there may be alternatives
You asked about Plugins. The result you want may be possible using a Skin. Skins, unlike Plugins, do not have any actual functionality, but they do replace the built-in art assets with custom art. I do not know whether the player position icon is exposed to Skins - perhaps you can send a private message to Adra here on lotrointerface.com, she will likely know if the player position art asset is accessible and whether any existing Skins update that art asset.
If this is not possible, then you may find the MoorMap plugin helpful. MoorMap displays an alternate map, not integrated with the built-in map directly. If you are just having trouble locating the player icon on the map, MoorMap has a "ping" effect that it displays at the player location on its map which makes finding the location easier. However, since player location and the built-in map are not exposed to Lua, MoorMap can not keep the player position updated automatically, the user has to press a button or enter the chat command "/loc" to update the position in MoorMap. MoorMap can not directly replace the in-game map because it can not display a quest tracker like the built-in map does (like the map itself, quest and deed data are not exposed to Lua). However, like the location ping, MoorMap has many features that are missing from the built-in map, such as custom annotations, resizing/repositioning, breadcrumbs, etc.