How do I prepare 2-(3-azido-3-oxopropoxy)ethyl methacrylate from methacrylic acid?
You're going need a protection strategy to make this work. Here is one possible route:

Start with 2-Chloro/bromo ethanol and protect the OH with a silyl
ether, $\ce{SiMe3}$ will do, but requires anhydrous workup.
React the protected haloethanol with the anion of
3-hydroxypropionate-t-butyl ester (commercially available) ($\ce{NaH,
NaI, DMF}$) then deprotect the product (mild acidic work up for $\ce{Me3SiO}$
or $\ce{F-}$ for others).
React this product with the acid chloride of methacrylic acid
(commercially available). Isolate and purify the product.
Remove the t-butyl ester with $\ce{TFA/CH2Cl2}$, concentrate to dry and
form the acyl chloride ($\ce{SOCl2}$, cat DMF).  React this with sodium
azide to give the product following the first part of this prep
(Org. Syn.).