Are there fields corresponding to the composite particles (e.g. hydrogen atom field)?
$\def\rr{{\bf r}}\def\ii{{\rm i}}\newcommand{\ket}[1]{\lvert#1\rangle}$I do mostly atomic physics (where, for example, the scattering of atoms is very pertinent) so I have a slightly different take on what counts as "useful" in QFT. It is often very useful indeed to define an "atomic field operator" $\Psi(\rr)$. The excitations of this field are entire atoms, which may be either bosonic or fermionic, in which case the fields satisfy appropriate (anti-)commutation relations $$\Psi(\rr)\Psi^\dagger(\rr') \pm \Psi^\dagger(\rr')\Psi(\rr) = \delta(\rr-\rr'),$$ where the minus (plus) sign is for bosons (fermions). Thus, if $\ket{0}$ is the vacuum state containing no particles, then a general $N$-particle state corresponds to $$\ket{\phi} = \int\prod_{j=1}^N\mathrm{d}\rr_j\; \phi(\rr_1,\rr_2,\ldots,\rr_N) \Psi^\dagger(\rr_1)\Psi^\dagger(\rr_2)\cdots \Psi^\dagger(\rr_N)\ket{0},$$ where $\phi(\rr_1,\ldots,\rr_N)$ is the $N$-body wave function in coordinate space.
So far we have considered the field operator in the Schroedinger picture. In the Heisenberg picture, the time evolution of the field is generated by the Heisenberg equation $$ \ii\hbar\dot{\Psi}(\rr,t) = -\frac{\hbar^2 \nabla^2}{2m} \Psi(\rr,t) + V[\Psi;\rr]\Psi(\rr,t),$$ where the (non-linear) potential reads as $$ V[\Psi;\rr] = V_1(\rr) + \int{\rm d}\rr'\; V_2(\rr-\rr') \Psi^\dagger(\rr',t)\Psi(\rr',t).$$ The first term $V_1(\rr)$ describes a one-particle external potential, the second term $V_2(\rr)$ describes a two-body interaction potential between the atoms. Higher-order (i.e. $n$-body potential) terms are possible as well. It is straightforward to generalise the field $\Psi(\rr)$ to a spinor having a component for each possible internal state of the atom. In this case the potential terms generalise to matrices or higher order tensors coupling together different internal states, while derivative terms in the potential could appear due to artificial gauge fields, for example.
As always in QFT, this is a low-energy effective theory. The non-relativistic approximation for the kinetic energy assumes that the centre-of-mass motion of the atoms is much slower than $c$. The field description of the atoms breaks down at scales where their internal structure becomes important, e.g. length scales comparable to the Bohr radius or energies comparable to the Rydberg energy.
If you want, you can view the above as the quantisation of a complex classical "Schroedinger" field $\Psi(\rr)$ (replace $\Psi^\dagger(\rr) \to \Psi^*(\rr)$). This classical Schroedinger field obeys a (non-linear) Schroedinger equation, with an associated Hamiltonian and Lagrangian etc. We usually do not do this, because the classical field does not describe anything particularly interesting. This is because 1) Nature is quantum and 2) no quantum states exist for which the classical field description is even close to a good approximation. The situation is different in, e.g., electrodynamics, where the observable dynamics of coherent states of the electromagnetic field can be well approximated by the classical Maxwell equations in many cases. However, in atomic field theories, the fundamental excitations are composed of fermions and their number is strictly conserved. This means that coherent states cannot be prepared* and therefore the classical field theory is largely useless (same reasoning applies, e.g. to the classical Dirac equation).
*In Bose-Einstein condensates, the off-diagonal long-range order makes a coherent state description a good approximation for many properties of interest. In this case the classical field theory is very useful; it goes by the name of the Gross-Pitaevskii equation.
It depends on the exact circumstance whether or not such an idea is a good approximation for the physics you want to describe.
For the hydrogen atom, you're usually not interesting in it's "scattering behaviour", you're interested in its internal energy states, how it behaves in external electromagnetic fields, etc. Such internal energy states are not well-modelled by QFT. In particular, you'll usually want to consider the proton as "fixed" and the electron as able to jump between its different energy levels. Considering the "hydrogen atom" as an indivisible (or atomic, as it were...) object is not particularly useful.
But there are composite particles where associating a field is perfectly sensible, for example the pion, whose effective field theory describes the nuclear force between hadrons - and the hadrons are also composite particles that are treated with a single field here, for instance by means of chiral perturbation theory.
There are, besides an interest in scattering behaviour (which you also might legitimately have for the hydrogen atom or other atoms, I'm not implying you should never treat the hydrogen atom this way), other reasons to model certain objects as the particles of a field:
Modern many body physics as in condensed matter theory is essentially quantum field theory, too, and it is very frequent there to have fields for composite particles, or even pseudo-particles like phonons. For instance, a simple but powerful model for superconductivity, the Landau model, just treats a conductor as a bunch of charged bosonic particles, thought of as the quanta of a field, coupled to the electromagnetic field, and superconductivity is then another instance of the Higgs mechanism of quantum field theory.
Every bound state (in particular every molecule and therefore the hydrogen atom) has an associated field, whose density figures in the statistical thermodynamics of chemical equilibrium and chemical reactions. Chemical reaction rates are computed microscopically in terms of the scattering theory of these fields or the corresponding asymptotic particles.
In nonrelativistic quantum mechanics, the corresponding field operators were constructed in Section 3 of the following paper:
W. Sandhas, Definition and existence of multichannel scattering states, Comm. Math. Phys. 3 (1966), 358-374. http://projecteuclid.org/download/pdf_1/euclid.cmp/1103839514
In the relativistic case, there is a corresponding nonperturbative construction in Haag-Ruelle scattering theory, valid at all energies (assuming the validity of the Wightman axioms). https://arxiv.org/abs/math-ph/0509047