There are 2 parts to your question.
The bomb part. I am assuming this means that there is a spontaneous reaction with an epicenter and a radially propagating explosion.
The endothermic part. This means that as the reaction progresses, it cools the surroundings as it passes.
Let us assume that such a reaction exists, which was spontaneous (had negative Gibbs energy) and endothermic:
G(p,T) = U + pV - TS \\
G(p,T) = H - TS
This would mean that as the reaction proceeds, the reactants and products would expand (given that the assumptions surrounding the explosion). For an endothermic reaction, $\Delta H$ is positive (as the internal energy rises $U$ increases from absorbing heat and the volume, $V$, is increasing for the fluid components of the reaction - from spreading out due to the explosion)
$T$ is decreasing over time. (as its an endothermic reaction)
$S$ is dependent on the nature of the reaction. In order for the reaction to be spontaneous at the start (our assumption for the start of the reaction), as the reaction went along, assuming that there mechanics of the reaction stayed the same (not true - will explain why later), this would still mean that the overall Gibbs energy would tend towards a positive value over time.
When it reaches 0, it stops being spontaneous and the reaction will stop being spontaneous. (it will propagate as long as it is kinetically permissible, i.e. activation constraints are satisfied)
Why I think $S$ will decrease over time
The reaction spreads radially so the reaction components themselves have to diffuse radially from the epicenter.
Considering the Boltzmann (stochastic, state-based) entropy as a measure of the system, in the system of the explosion, the fluid components' internal energy only decreases. This means that the overall disorder of the system decreases - thereby decreasing the entropy. The gaseous components dominate the measurement of entropy therefore, this means that there $S$ will decrease over time.
An answer but not to your question
Drop a balloon filled with liquid nitrogen and it will be close to what I think you're imagining.
This isn't really a 'reaction' but will have the effect of an endothermic bomb (the shockblast will be from the expanding nitrogen) and the surroundings will be cooled due to nitrogen absorbing the latent heat.
How about a hypothetical endothermic self-replicating nanothermitic reaction?
The principle behind a self-replicating nanothermitic reaction is that the the reaction components produce the feed stock required for the reaction to continue indefinitely from the surroundings.
The absorbed energy would provide a access to the high-energy quantum states required to pass the activation energy barrier (of this hypothetical reaction pathway) and the solid products emitted would be left at such a low temperature that they supercool their surroundings as they pass through.
This is just food for thought, but I guess but I don't think that the reaction conditions on Earth can sustain such a reaction.