Many variables of interest in agricultural or economical surveys have skewed distributions and can be contaminated with a non-negligible portion of zeros. A small area estimation methodology is developed based on the assumption that the responses given covariates follows a zero-inflated lognormal model with correlated random area effects between the positive part and the binary part. Existing literature tends to assume independence between the two parts or adopts Bayesian methods for inference. We prefer to adopt a frequentist approach since we lack prior information that would guide an appropriate choice of prior distributions and constructing a posterior distribution for all elements of the large population of cropland in South Dakota is computationally expensive. We address the analytical challenges arising in the frequentist analysis under the model assumption through our development of maximum likelihood estimators and empirical Bayes (EB) predictors. A simulation study shows (1) the proposed EB predictor works reasonably well; (2) ignoring the correlation between the two parts may lead to efficiency loss in terms of MSE. The methodology is applied to estimate sheet and rill erosion rates for counties in South Dakota using data from the Conservation Effects Assessment Project (CEAP) and auxiliary information collected from the Soil Survey Geographic Data (SSURGO) and Cropland Data Layer (CDL).