= Solution
Use the <hypercharge> normalization $Q_{\rm electric}=T^3+Y$. All matter <chiral superfields> are written with left-handed <Weyl spinors>, so the fields denoted by a superscript $c$ contain the charge conjugates of the usual right-handed <Standard Model fermions>. The <MSSM superfield representations> are
$$
\begin{array}{c|c|c}
\text{chiral superfield}&SU(3)_c\times SU(2)_L\times U(1)_Y&\text{multiplicity}\\\hline
Q&(\mathbf3,\mathbf2,1/6)&3\\
U^c&(\overline{\mathbf3},\mathbf1,-2/3)&3\\
D^c&(\overline{\mathbf3},\mathbf1,1/3)&3\\
L&(\mathbf1,\mathbf2,-1/2)&3\\
E^c&(\mathbf1,\mathbf1,1)&3\\
H_u&(\mathbf1,\mathbf2,1/2)&1\\
H_d&(\mathbf1,\mathbf2,-1/2)&1
\end{array}
$$
Each <fermion generation> contributes the first five <chiral superfields>. Their <complex scalar field> partners are <squarks> for the <quarks> and <sleptons> for the <leptons>. Each <Higgs chiral doublet> contains a Higgs <complex scalar field> and a <higgsino>. Every <chiral superfield> also has a complex <auxiliary field>. The <vector superfields> are
$$
\boxed{V_3:(\mathbf8,\mathbf1,0),\qquad V_2:(\mathbf1,\mathbf3,0),\qquad V_1:(\mathbf1,\mathbf1,0).}
$$
They contain the corresponding <gauge bosons>, <gauginos> in the <Adjoint representation>, and real <auxiliary fields>. A right-handed neutrino <chiral superfield> is not part of the minimal field content.
A <gauge anomaly> is a quantum obstruction to a classical <gauge symmetry>. For <hypercharge>, triangle diagrams with left-handed <Weyl spinors> can violate the Ward identity of the <gauge boson>; an uncancelled <gauge anomaly> makes the <gauge theory> inconsistent. <Anomaly cancellation> sums over every component, including colour and weak multiplicities. <Complex scalar fields> do not contribute to these chiral <gauge anomalies>. For one <fermion generation>, the cubic <hypercharge> coefficient is
$$
\begin{aligned}
\mathcal A_{Y^3}
&=6(1/6)^3+3(-2/3)^3+3(1/3)^3+2(-1/2)^3+1^3\\
&=\frac1{36}-\frac89+\frac19-\frac14+1=\boxed{0}.
\end{aligned}
$$
The other coefficients involving a <hypercharge> <gauge boson> vanish too. With the fundamental index $T(\mathbf N)=1/2$,
$$
\begin{aligned}
\mathcal A_{SU(3)^2Y}&=2(1/2)(1/6)+(1/2)(-2/3)+(1/2)(1/3)=0,\\
\mathcal A_{SU(2)^2Y}&=3(1/2)(1/6)+(1/2)(-1/2)=0,\\
\mathcal A_{\mathrm{grav}^2Y}&=6(1/6)+3(-2/3)+3(1/3)+2(-1/2)+1=0.
\end{aligned}
$$
The last line is the <mixed gauge-gravitational anomaly>. Coefficients with one non-Abelian generator and two <hypercharge> generators vanish by tracelessness. For completeness, the purely colour cubic <gauge anomaly> cancels between the two fundamental quark components and the two antifundamentals; the weak group has no perturbative cubic <gauge anomaly>. Its four left-handed doublets per <fermion generation> also avoid the <Witten SU(2) anomaly>. Thus \b[each family is separately anomaly-free], not merely their sum.
The <gauginos> do not spoil this result: their <hypercharge> is zero and their <Adjoint representation> is real. One extra <Higgs chiral doublet> is different because its <higgsino> is chiral. For $H_u$, its contributions are
$$
\mathcal A_{Y^3}=2(1/2)^3=1/4,\qquad
\mathcal A_{SU(2)^2Y}=(1/2)(1/2)=1/4,\qquad
\mathcal A_{\mathrm{grav}^2Y}=2(1/2)=1.
$$
They have no compensating contribution from the Higgs <complex scalar field>. The <higgsino> in $H_d$ supplies precisely the negative of each coefficient. \b[Opposite-hypercharge Higgs chiral doublets restore anomaly cancellation.] The same pair restores an even number of weak fermion doublets, so the <Witten SU(2) anomaly> provides an additional check of the <higgsino anomaly cancellation>.
The independent reason is the <holomorphic need for two Higgs chiral doublets>. A <superpotential> is a <holomorphic function> of <chiral superfields>, so it cannot use a conjugate Higgs <superfield> to generate the missing <Yukawa couplings>. The ordinary <Standard Model> can use a Higgs scalar and its conjugate, but the <MSSM> needs distinct <chiral superfields> of both <hypercharges>. For example,
$$
W_{\rm Yukawa}=y_u^{ij}U_i^c Q_j\mathbin{\cdot}H_u-y_d^{ij}D_i^c Q_j\mathbin{\cdot}H_d-y_e^{ij}E_i^c L_j\mathbin{\cdot}H_d,
$$
where the dot contracts weak indices with the antisymmetric tensor. All three terms are gauge-invariant <holomorphic functions>. \b[$H_u$ supplies up-type masses, while $H_d$ supplies down-type and charged-lepton masses.] Replacing either by the conjugate of the other would violate the <holomorphic closure of chiral superfields>.
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