Current optical vegetation indices (VIs) for monitoring forest ecosystems are well established and widely used in various applications, but can be limited by atmospheric effects such as clouds. In contrast, synthetic aperture radar (SAR) data can offer insightful and systematic forest monitoring with complete time series (TS) due to signal penetration through clouds and day and night image acquisitions. This study aims to address the limitations of optical satellite data by using SAR data as an alternative for estimating optical VIs for forests through machine learning (ML). While this approach is less direct and likely only feasible through the power of ML, it raises the scientific question of whether enough relevant information is contained in the SAR signal to accurately estimate VIs. This work covers the estimation of TS of four VIs (LAI, FAPAR, EVI and NDVI) using multitemporal Sentinel-1 SAR and ancillary data. The study focused on both healthy and disturbed temperate forest areas in Czechia for the year 2021, while ground truth labels generated from Sentinel-2 multispectral data. This was enabled by creating a paired multi-modal TS dataset in Google Earth Engine (GEE), including temporally and spatially aligned Sentinel-1, Sentinel-2, DEM, weather and land cover datasets. The inclusion of DEM-derived auxiliary features and additional meteorological information, further improved the results. In the comparison of ML models, the traditional ML algorithms, RFR and XGBoost slightly outperformed the AutoML approach, auto-sklearn, for all VIs, achieving high accuracies ($R^2$ between 70-86%) and low errors (0.055-0.29 of MAE). In general, up to 240 measurements per year and a spatial resolution of 20 m can be achieved using estimated SAR-based VIs with high accuracy. A great advantage of the SAR-based VI is the ability to detect abrupt forest changes with sub-weekly temporal accuracy.