Can I use some type of transistor i.e. BJT or MOSFET to act as normally closed relay?
Yes I could hold the Arduino pin high instead but this would waste more power.

That's a wrong assumption. It is NOT TRUE. Microprocessors like the ATMega328 that is used in many Arduino modules are build using CMOS technology and that means that they can maintain a voltage (low or high) at any output and as long as no current flows then this does not consume any power.
If you would use a MOSFET (connected to the output of the Arduino) to switch on/off your relay or LED then no current would flow as a MOSFET uses similar technology so again, no current flows to keep the MOSFET on or off.
So no power is used to keep a MOSFET continuously in an on or off state. Only when it is switched on =&gt; off or off =&gt; on a little bit of power is needed.
So your question is based on a false premise.
Yes, normally of devices do exist: depletion type MOSFETs and JFETs as well.
But these aren't very suitable for switching things they're used in some niche applications for amplifying the voltage of an electret microphone or amplifying RF signals.
TLDR: you're asking for a device that's unneeded as we can solve the &quot;problems&quot; you see using simple circuitry. These are common solutions that all designers know about.